NAME

complexHEcomputational - complex

SYNOPSIS

Functions


subroutine checon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
 
CHECON subroutine checon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO)
 
CHECON_3 subroutine checon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
 
CHECON_ROOK estimates the reciprocal of the condition number fort HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges) subroutine cheequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
 
CHEEQUB subroutine chegs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
 
CHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the factorization results obtained from cpotrf (unblocked algorithm). subroutine chegst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
 
CHEGST subroutine cherfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
 
CHERFS subroutine cherfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
 
CHERFSX subroutine chetd2 (UPLO, N, A, LDA, D, E, TAU, INFO)
 
CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm). subroutine chetf2 (UPLO, N, A, LDA, IPIV, INFO)
 
CHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS). subroutine chetf2_rk (UPLO, N, A, LDA, E, IPIV, INFO)
 
CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). subroutine chetf2_rook (UPLO, N, A, LDA, IPIV, INFO)
 
CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). subroutine chetrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
 
CHETRD subroutine chetrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
 
CHETRD_2STAGE subroutine chetrd_he2hb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
 
CHETRD_HE2HB subroutine chetrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
 
CHETRF subroutine chetrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
 
CHETRF_AA subroutine chetrf_rk (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
 
CHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). subroutine chetrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
 
CHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (blocked algorithm, calling Level 3 BLAS). subroutine chetri (UPLO, N, A, LDA, IPIV, WORK, INFO)
 
CHETRI subroutine chetri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
 
CHETRI2 subroutine chetri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO)
 
CHETRI2X subroutine chetri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
 
CHETRI_3 subroutine chetri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO)
 
CHETRI_3X subroutine chetri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO)
 
CHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method. subroutine chetrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
 
CHETRS subroutine chetrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)
 
CHETRS2 subroutine chetrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
 
CHETRS_3 subroutine chetrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)
 
CHETRS_AA subroutine chetrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
 
CHETRS_ROOK computes the solution to a system of linear equations A * X = B for HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges) subroutine cla_heamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
 
CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds. real function cla_hercond_c (UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK)
 
CLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices. real function cla_hercond_x (UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)
 
CLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices. subroutine cla_herfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
 
CLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. real function cla_herpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
 
CLA_HERPVGRW subroutine clahef (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
 
CLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS). subroutine clahef_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)
 
CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. subroutine clahef_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
 

Detailed Description

This is the group of complex computational functions for HE matrices

Function Documentation

subroutine checon (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

CHECON
Purpose:
 
 CHECON estimates the reciprocal of the condition number of a complex
 Hermitian matrix A using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHETRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHETRF.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF.
ANORM
          ANORM is REAL
          The 1-norm of the original matrix A.
RCOND
          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.
WORK
          WORK is COMPLEX array, dimension (2*N)
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine checon_3 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

CHECON_3
Purpose:
 
 CHECON_3 estimates the reciprocal of the condition number (in the
 1-norm) of a complex Hermitian matrix A using the factorization
 computed by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). This routine uses BLAS3 solver CHETRS_3.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are
          stored as an upper or lower triangular matrix:
          = 'U':  Upper triangular, form is A = P*U*D*(U**H)*(P**T);
          = 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          Diagonal of the block diagonal matrix D and factors U or L
          as computed by CHETRF_RK and CHETRF_BK:
            a) ONLY diagonal elements of the Hermitian block diagonal
               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
               (superdiagonal (or subdiagonal) elements of D
                should be provided on entry in array E), and
            b) If UPLO = 'U': factor U in the superdiagonal part of A.
               If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On entry, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
          If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_RK or CHETRF_BK.
ANORM
          ANORM is REAL
          The 1-norm of the original matrix A.
RCOND
          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.
WORK
          WORK is COMPLEX array, dimension (2*N)
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  June 2017,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine checon_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

CHECON_ROOK estimates the reciprocal of the condition number fort HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges)
Purpose:
 
 CHECON_ROOK estimates the reciprocal of the condition number of a complex
 Hermitian matrix A using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHETRF_ROOK.
An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHETRF_ROOK.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_ROOK.
ANORM
          ANORM is REAL
          The 1-norm of the original matrix A.
RCOND
          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.
WORK
          WORK is COMPLEX array, dimension (2*N)
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine cheequb (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, complex, dimension( * ) WORK, integer INFO)

CHEEQUB
Purpose:
 
 CHEEQUB computes row and column scalings intended to equilibrate a
 Hermitian matrix A (with respect to the Euclidean norm) and reduce
 its condition number. The scale factors S are computed by the BIN
 algorithm (see references) so that the scaled matrix B with elements
 B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
 the smallest possible condition number over all possible diagonal
 scalings.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A. N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The N-by-N Hermitian matrix whose scaling factors are to be
          computed.
LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).
S
          S is REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.
SCOND
          SCOND is REAL
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i). If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.
AMAX
          AMAX is REAL
          Largest absolute value of any matrix element. If AMAX is
          very close to overflow or very close to underflow, the
          matrix should be scaled.
WORK
          WORK is COMPLEX array, dimension (2*N)
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
References:
Livne, O.E. and Golub, G.H., 'Scaling by Binormalization',
 

Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
 

DOI 10.1023/B:NUMA.0000016606.32820.69
 

Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

subroutine chegs2 (integer ITYPE, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the factorization results obtained from cpotrf (unblocked algorithm).
Purpose:
 
 CHEGS2 reduces a complex Hermitian-definite generalized
 eigenproblem to standard form.
If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H *A*L.
B must have been previously factorized as U**H *U or L*L**H by ZPOTRF.
Parameters
ITYPE
          ITYPE is INTEGER
          = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
          = 2 or 3: compute U*A*U**H or L**H *A*L.
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored, and how B has been factorized.
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrices A and B.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the same format as A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
B
          B is COMPLEX array, dimension (LDB,N)
          The triangular factor from the Cholesky factorization of B,
          as returned by CPOTRF.
          B is modified by the routine but restored on exit.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chegst (integer ITYPE, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHEGST
Purpose:
 
 CHEGST reduces a complex Hermitian-definite generalized
 eigenproblem to standard form.
If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H*A*L.
B must have been previously factorized as U**H*U or L*L**H by CPOTRF.
Parameters
ITYPE
          ITYPE is INTEGER
          = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
          = 2 or 3: compute U*A*U**H or L**H*A*L.
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**H*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**H.
N
          N is INTEGER
          The order of the matrices A and B.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the same format as A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
B
          B is COMPLEX array, dimension (LDB,N)
          The triangular factor from the Cholesky factorization of B,
          as returned by CPOTRF.
          B is modified by the routine but restored on exit.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine cherfs (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CHERFS
Purpose:
 
 CHERFS improves the computed solution to a system of linear
 equations when the coefficient matrix is Hermitian indefinite, and
 provides error bounds and backward error estimates for the solution.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of A contains the upper triangular part
          of the matrix A, and the strictly lower triangular part of A
          is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of
          the matrix A, and the strictly upper triangular part of A is
          not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
AF
          AF is COMPLEX array, dimension (LDAF,N)
          The factored form of the matrix A.  AF contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**H or
          A = L*D*L**H as computed by CHETRF.
LDAF
          LDAF is INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
X
          X is COMPLEX array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by CHETRS.
          On exit, the improved solution matrix X.
LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
FERR
          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.
BERR
          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).
WORK
          WORK is COMPLEX array, dimension (2*N)
RWORK
          RWORK is REAL array, dimension (N)
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
 
  ITMAX is the maximum number of steps of iterative refinement.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine cherfsx (character UPLO, character EQUED, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) S, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CHERFSX
Purpose:
 
    CHERFSX improves the computed solution to a system of linear
    equations when the coefficient matrix is Hermitian indefinite, and
    provides error bounds and backward error estimates for the
    solution.  In addition to normwise error bound, the code provides
    maximum componentwise error bound if possible.  See comments for
    ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In this case, the solution and error bounds returned are for the original unequilibrated system.
     Some optional parameters are bundled in the PARAMS array.  These
     settings determine how refinement is performed, but often the
     defaults are acceptable.  If the defaults are acceptable, users
     can pass NPARAMS = 0 which prevents the source code from accessing
     the PARAMS argument.
Parameters
UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
EQUED
          EQUED is CHARACTER*1
     Specifies the form of equilibration that was done to A
     before calling this routine. This is needed to compute
     the solution and error bounds correctly.
       = 'N':  No equilibration
       = 'Y':  Both row and column equilibration, i.e., A has been
               replaced by diag(S) * A * diag(S).
               The right hand side B has been changed accordingly.
N
          N is INTEGER
     The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
     The number of right hand sides, i.e., the number of columns
     of the matrices B and X.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
     The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N
     upper triangular part of A contains the upper triangular
     part of the matrix A, and the strictly lower triangular
     part of A is not referenced.  If UPLO = 'L', the leading
     N-by-N lower triangular part of A contains the lower
     triangular part of the matrix A, and the strictly upper
     triangular part of A is not referenced.
LDA
          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).
AF
          AF is COMPLEX array, dimension (LDAF,N)
     The factored form of the matrix A.  AF contains the block
     diagonal matrix D and the multipliers used to obtain the
     factor U or L from the factorization A = U*D*U**H or A =
     L*D*L**H as computed by CHETRF.
LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
     Details of the interchanges and the block structure of D
     as determined by CHETRF.
S
          S is REAL array, dimension (N)
     The scale factors for A.  If EQUED = 'Y', A is multiplied on
     the left and right by diag(S).  S is an input argument if FACT =
     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
     = 'Y', each element of S must be positive.  If S is output, each
     element of S is a power of the radix. If S is input, each element
     of S should be a power of the radix to ensure a reliable solution
     and error estimates. Scaling by powers of the radix does not cause
     rounding errors unless the result underflows or overflows.
     Rounding errors during scaling lead to refining with a matrix that
     is not equivalent to the input matrix, producing error estimates
     that may not be reliable.
B
          B is COMPLEX array, dimension (LDB,NRHS)
     The right hand side matrix B.
LDB
          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).
X
          X is COMPLEX array, dimension (LDX,NRHS)
     On entry, the solution matrix X, as computed by CHETRS.
     On exit, the improved solution matrix X.
LDX
          LDX is INTEGER
     The leading dimension of the array X.  LDX >= max(1,N).
RCOND
          RCOND is REAL
     Reciprocal scaled condition number.  This is an estimate of the
     reciprocal Skeel condition number of the matrix A after
     equilibration (if done).  If this is less than the machine
     precision (in particular, if it is zero), the matrix is singular
     to working precision.  Note that the error may still be small even
     if this number is very small and the matrix appears ill-
     conditioned.
BERR
          BERR is REAL array, dimension (NRHS)
     Componentwise relative backward error.  This is the
     componentwise relative backward error of each solution vector X(j)
     (i.e., the smallest relative change in any element of A or B that
     makes X(j) an exact solution).
N_ERR_BNDS
          N_ERR_BNDS is INTEGER
     Number of error bounds to return for each right hand side
     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
     ERR_BNDS_COMP below.
ERR_BNDS_NORM
          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i))
The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra cautions.
ERR_BNDS_COMP
          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i))
The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra cautions.
NPARAMS
          NPARAMS is INTEGER
     Specifies the number of parameters set in PARAMS.  If <= 0, the
     PARAMS array is never referenced and default values are used.
PARAMS
          PARAMS is REAL array, dimension NPARAMS
     Specifies algorithm parameters.  If an entry is < 0.0, then
     that entry will be filled with default value used for that
     parameter.  Only positions up to NPARAMS are accessed; defaults
     are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not. Default: 1.0 = 0.0: No refinement is performed, and no error bounds are computed. = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. (other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement. Default: 10 Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)
WORK
          WORK is COMPLEX array, dimension (2*N)
RWORK
          RWORK is REAL array, dimension (2*N)
INFO
          INFO is INTEGER
       = 0:  Successful exit. The solution to every right-hand side is
         guaranteed.
       < 0:  If INFO = -i, the i-th argument had an illegal value
       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
         has been completed, but the factor U is exactly singular, so
         the solution and error bounds could not be computed. RCOND = 0
         is returned.
       = N+J: The solution corresponding to the Jth right-hand side is
         not guaranteed. The solutions corresponding to other right-
         hand sides K with K > J may not be guaranteed as well, but
         only the first such right-hand side is reported. If a small
         componentwise error is not requested (PARAMS(3) = 0.0) then
         the Jth right-hand side is the first with a normwise error
         bound that is not guaranteed (the smallest J such
         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
         the Jth right-hand side is the first with either a normwise or
         componentwise error bound that is not guaranteed (the smallest
         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
         about all of the right-hand sides check ERR_BNDS_NORM or
         ERR_BNDS_COMP.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetd2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, integer INFO)

CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
Purpose:
 
 CHETD2 reduces a complex Hermitian matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q**H * A * Q = T.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the unitary matrix Q as a product
          of elementary reflectors. See Further Details.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
D
          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).
E
          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU
          TAU is COMPLEX array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).
The contents of A on exit are illustrated by the following examples with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

subroutine chetf2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

CHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS).
Purpose:
 
 CHETF2 computes the factorization of a complex Hermitian matrix A
 using the Bunch-Kaufman diagonal pivoting method:
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  09-29-06 - patch from
    Bobby Cheng, MathWorks
Replace l.210 and l.392 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN by IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
If UPLO = 'U', then A = U*D*U**H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

subroutine chetf2_rk (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)

CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
 
 CHETF2_RK computes the factorization of a complex Hermitian matrix A
 using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS. For more information see Further Details section.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.
            If UPLO = 'U': the leading N-by-N upper triangular part
            of A contains the upper triangular part of the matrix A,
            and the strictly lower triangular part of A is not
            referenced.
If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
On exit, contains: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On exit, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          IPIV describes the permutation matrix P in the factorization
          of matrix A as follows. The absolute value of IPIV(k)
          represents the index of row and column that were
          interchanged with the k-th row and column. The value of UPLO
          describes the order in which the interchanges were applied.
          Also, the sign of IPIV represents the block structure of
          the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
          diagonal blocks which correspond to 1 or 2 interchanges
          at each factorization step. For more info see Further
          Details section.
If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
          INFO is INTEGER
          = 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
 TODO: put further details
Contributors:
 
  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

subroutine chetf2_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
Purpose:
 
 CHETF2_ROOK computes the factorization of a complex Hermitian matrix A
 using the bounded Bunch-Kaufman ('rook') diagonal pivoting method:
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  If UPLO = 'U', then A = U*D*U**H, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
Contributors:
 
  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

subroutine chetrd (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRD
Purpose:
 
 CHETRD reduces a complex Hermitian matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q**H * A * Q = T.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the unitary matrix Q as a product
          of elementary reflectors. See Further Details.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
D
          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).
E
          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU
          TAU is COMPLEX array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).
WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
          LWORK is INTEGER
          The dimension of the array WORK.  LWORK >= 1.
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).
The contents of A on exit are illustrated by the following examples with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

subroutine chetrd_2stage (character VECT, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( * ) HOUS2, integer LHOUS2, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRD_2STAGE
Purpose:
 
 CHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q1**H Q2**H* A * Q2 * Q1 = T.
Parameters
VECT
          VECT is CHARACTER*1
          = 'N':  No need for the Housholder representation, 
                  in particular for the second stage (Band to
                  tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
          = 'V':  the Householder representation is needed to 
                  either generate Q1 Q2 or to apply Q1 Q2, 
                  then LHOUS2 is to be queried and computed.
                  (NOT AVAILABLE IN THIS RELEASE).
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the band superdiagonal
          of A are overwritten by the corresponding elements of the
          internal band-diagonal matrix AB, and the elements above 
          the KD superdiagonal, with the array TAU, represent the unitary
          matrix Q1 as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and band subdiagonal of A are over-
          written by the corresponding elements of the internal band-diagonal
          matrix AB, and the elements below the KD subdiagonal, with
          the array TAU, represent the unitary matrix Q1 as a product
          of elementary reflectors. See Further Details.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
D
          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.
E
          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T.
TAU
          TAU is COMPLEX array, dimension (N-KD)
          The scalar factors of the elementary reflectors of 
          the first stage (see Further Details).
HOUS2
          HOUS2 is COMPLEX array, dimension (LHOUS2)
          Stores the Householder representation of the stage2
          band to tridiagonal.
LHOUS2
          LHOUS2 is INTEGER
          The dimension of the array HOUS2.
          If LWORK = -1, or LHOUS2=-1,
          then a query is assumed; the routine
          only calculates the optimal size of the HOUS2 array, returns
          this value as the first entry of the HOUS2 array, and no error
          message related to LHOUS2 is issued by XERBLA.
          If VECT='N', LHOUS2 = max(1, 4*n);
          if VECT='V', option not yet available.
WORK
          WORK is COMPLEX array, dimension (LWORK)
LWORK
          LWORK is INTEGER
          The dimension of the array WORK. LWORK = MAX(1, dimension)
          If LWORK = -1, or LHOUS2 = -1,
          then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
          LWORK = MAX(1, dimension) where
          dimension   = max(stage1,stage2) + (KD+1)*N
                      = N*KD + N*max(KD+1,FACTOPTNB) 
                        + max(2*KD*KD, KD*NTHREADS) 
                        + (KD+1)*N 
          where KD is the blocking size of the reduction,
          FACTOPTNB is the blocking used by the QR or LQ
          algorithm, usually FACTOPTNB=128 is a good choice
          NTHREADS is the number of threads used when
          openMP compilation is enabled, otherwise =1.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  Implemented by Azzam Haidar.
All details are available on technical report, SC11, SC13 papers.
Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

subroutine chetrd_he2hb (character UPLO, integer N, integer KD, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRD_HE2HB
Purpose:
 
 CHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
 band-diagonal form AB by a unitary similarity transformation:
 Q**H * A * Q = AB.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
KD
          KD is INTEGER
          The number of superdiagonals of the reduced matrix if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
          The reduced matrix is stored in the array AB.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the unitary matrix Q as a product
          of elementary reflectors. See Further Details.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
AB
          AB is COMPLEX array, dimension (LDAB,N)
          On exit, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.
TAU
          TAU is COMPLEX array, dimension (N-KD)
          The scalar factors of the elementary reflectors (see Further
          Details).
WORK
          WORK is COMPLEX array, dimension (LWORK)
          On exit, if INFO = 0, or if LWORK=-1, 
          WORK(1) returns the size of LWORK.
LWORK
          LWORK is INTEGER
          The dimension of the array WORK which should be calculated
          by a workspace query. LWORK = MAX(1, LWORK_QUERY)
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
          where FACTOPTNB is the blocking used by the QR or LQ
          algorithm, usually FACTOPTNB=128 is a good choice otherwise
          putting LWORK=-1 will provide the size of WORK.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  Implemented by Azzam Haidar.
All details are available on technical report, SC11, SC13 papers.
Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = n-kd.
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i).
The contents of A on exit are illustrated by the following examples with n = 5:
if UPLO = 'U': if UPLO = 'L':
( ab ab/v1 v1 v1 v1 ) ( ab ) ( ab ab/v2 v2 v2 ) ( ab/v1 ab ) ( ab ab/v3 v3 ) ( v1 ab/v2 ab ) ( ab ab/v4 ) ( v1 v2 ab/v3 ab ) ( ab ) ( v1 v2 v3 ab/v4 ab )
where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)..fi
 

subroutine chetrf (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRF
Purpose:
 
 CHETRF computes the factorization of a complex Hermitian matrix A
 using the Bunch-Kaufman diagonal pivoting method.  The form of the
 factorization is
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
          LWORK is INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned by ILAENV.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                has been completed, but the block diagonal matrix D is
                exactly singular, and division by zero will occur if it
                is used to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  If UPLO = 'U', then A = U*D*U**H, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

subroutine chetrf_aa (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRF_AA
Purpose:
 
 CHETRF_AA computes the factorization of a complex hermitian matrix A
 using the Aasen's algorithm.  The form of the factorization is
A = U**H*T*U or A = L*T*L**H
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a hermitian tridiagonal matrix.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, the tridiagonal matrix is stored in the diagonals and the subdiagonals of A just below (or above) the diagonals, and L is stored below (or above) the subdiaonals, when UPLO is 'L' (or 'U').
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          On exit, it contains the details of the interchanges, i.e.,
          the row and column k of A were interchanged with the
          row and column IPIV(k).
WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
          LWORK is INTEGER
          The length of WORK.  LWORK >= 2*N. For optimum performance
          LWORK >= N*(1+NB), where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetrf_rk (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
Purpose:
 
 CHETRF_RK computes the factorization of a complex Hermitian matrix A
 using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.
            If UPLO = 'U': the leading N-by-N upper triangular part
            of A contains the upper triangular part of the matrix A,
            and the strictly lower triangular part of A is not
            referenced.
If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
On exit, contains: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On exit, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          IPIV describes the permutation matrix P in the factorization
          of matrix A as follows. The absolute value of IPIV(k)
          represents the index of row and column that were
          interchanged with the k-th row and column. The value of UPLO
          describes the order in which the interchanges were applied.
          Also, the sign of IPIV represents the block structure of
          the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
          diagonal blocks which correspond to 1 or 2 interchanges
          at each factorization step. For more info see Further
          Details section.
If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
WORK
          WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
          LWORK is INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned
          by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
 TODO: put correct description
Contributors:
 
  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine chetrf_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
Purpose:
 
 CHETRF_ROOK computes the factorization of a complex Hermitian matrix A
 using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
 The form of the factorization is
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
If UPLO = 'U': Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L': Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK)).
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
          LWORK is INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                has been completed, but the block diagonal matrix D is
                exactly singular, and division by zero will occur if it
                is used to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
Contributors:
 
  June 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine chetri (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer INFO)

CHETRI
Purpose:
 
 CHETRI computes the inverse of a complex Hermitian indefinite matrix
 A using the factorization A = U*D*U**H or A = L*D*L**H computed by
 CHETRF.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by CHETRF.
On exit, if INFO = 0, the (Hermitian) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF.
WORK
          WORK is COMPLEX array, dimension (N)
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetri2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRI2
Purpose:
 
 CHETRI2 computes the inverse of a COMPLEX hermitian indefinite matrix
 A using the factorization A = U*D*U**T or A = L*D*L**T computed by
 CHETRF. CHETRI2 set the LEADING DIMENSION of the workspace
 before calling CHETRI2X that actually computes the inverse.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by CHETRF.
On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF.
WORK
          WORK is COMPLEX array, dimension (N+NB+1)*(NB+3)
LWORK
          LWORK is INTEGER
          The dimension of the array WORK.
          WORK is size >= (N+NB+1)*(NB+3)
          If LWORK = -1, then a workspace query is assumed; the routine
           calculates:
              - the optimal size of the WORK array, returns
          this value as the first entry of the WORK array,
              - and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetri2x (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( n+nb+1,* ) WORK, integer NB, integer INFO)

CHETRI2X
Purpose:
 
 CHETRI2X computes the inverse of a complex Hermitian indefinite matrix
 A using the factorization A = U*D*U**H or A = L*D*L**H computed by
 CHETRF.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the NNB diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by CHETRF.
On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the NNB structure of D
          as determined by CHETRF.
WORK
          WORK is COMPLEX array, dimension (N+NB+1,NB+3)
NB
          NB is INTEGER
          Block size
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetri_3 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRI_3
Purpose:
 
 CHETRI_3 computes the inverse of a complex Hermitian indefinite
 matrix A using the factorization computed by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
CHETRI_3 sets the leading dimension of the workspace before calling CHETRI_3X that actually computes the inverse. This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are
          stored as an upper or lower triangular matrix.
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, diagonal of the block diagonal matrix D and
          factors U or L as computed by CHETRF_RK and CHETRF_BK:
            a) ONLY diagonal elements of the Hermitian block diagonal
               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
               (superdiagonal (or subdiagonal) elements of D
                should be provided on entry in array E), and
            b) If UPLO = 'U': factor U in the superdiagonal part of A.
               If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, if INFO = 0, the Hermitian inverse of the original matrix. If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On entry, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
          If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_RK or CHETRF_BK.
WORK
          WORK is COMPLEX array, dimension (N+NB+1)*(NB+3).
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
          LWORK is INTEGER
          The length of WORK. LWORK >= (N+NB+1)*(NB+3).
If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  November 2017,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley

subroutine chetri_3x (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)

CHETRI_3X
Purpose:
 
 CHETRI_3X computes the inverse of a complex Hermitian indefinite
 matrix A using the factorization computed by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are
          stored as an upper or lower triangular matrix.
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, diagonal of the block diagonal matrix D and
          factors U or L as computed by CHETRF_RK and CHETRF_BK:
            a) ONLY diagonal elements of the Hermitian block diagonal
               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
               (superdiagonal (or subdiagonal) elements of D
                should be provided on entry in array E), and
            b) If UPLO = 'U': factor U in the superdiagonal part of A.
               If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, if INFO = 0, the Hermitian inverse of the original matrix. If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On entry, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_RK or CHETRF_BK.
WORK
          WORK is COMPLEX array, dimension (N+NB+1,NB+3).
NB
          NB is INTEGER
          Block size.
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  June 2017,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley

subroutine chetri_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer INFO)

CHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
Purpose:
 
 CHETRI_ROOK computes the inverse of a complex Hermitian indefinite matrix
 A using the factorization A = U*D*U**H or A = L*D*L**H computed by
 CHETRF_ROOK.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by CHETRF_ROOK.
On exit, if INFO = 0, the (Hermitian) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_ROOK.
WORK
          WORK is COMPLEX array, dimension (N)
INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine chetrs (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHETRS
Purpose:
 
 CHETRS solves a system of linear equations A*X = B with a complex
 Hermitian matrix A using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHETRF.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHETRF.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetrs2 (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integer INFO)

CHETRS2
Purpose:
 
 CHETRS2 solves a system of linear equations A*X = B with a complex
 Hermitian matrix A using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHETRF and converted by CSYCONV.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHETRF.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
WORK
          WORK is COMPLEX array, dimension (N)
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetrs_3 (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHETRS_3
Purpose:
 
 CHETRS_3 solves a system of linear equations A * X = B with a complex
 Hermitian matrix A using the factorization computed
 by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This algorithm is using Level 3 BLAS.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are
          stored as an upper or lower triangular matrix:
          = 'U':  Upper triangular, form is A = P*U*D*(U**H)*(P**T);
          = 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          Diagonal of the block diagonal matrix D and factors U or L
          as computed by CHETRF_RK and CHETRF_BK:
            a) ONLY diagonal elements of the Hermitian block diagonal
               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
               (superdiagonal (or subdiagonal) elements of D
                should be provided on entry in array E), and
            b) If UPLO = 'U': factor U in the superdiagonal part of A.
               If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On entry, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
          If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_RK or CHETRF_BK.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  June 2017,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine chetrs_aa (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHETRS_AA
Purpose:
 
 CHETRS_AA solves a system of linear equations A*X = B with a complex
 hermitian matrix A using the factorization A = U**H*T*U or
 A = L*T*L**H computed by CHETRF_AA.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U**H*T*U;
          = 'L':  Lower triangular, form is A = L*T*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          Details of factors computed by CHETRF_AA.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges as computed by CHETRF_AA.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
LWORK
          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,3*N-2).
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine chetrs_rook (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHETRS_ROOK computes the solution to a system of linear equations A * X = B for HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges)
Purpose:
 
 CHETRS_ROOK solves a system of linear equations A*X = B with a complex
 Hermitian matrix A using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHETRF_ROOK.
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHETRF_ROOK.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHETRF_ROOK.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.
LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine cla_heamv (integer UPLO, integer N, real ALPHA, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)

CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.
Purpose:
 
 CLA_SYAMV  performs the matrix-vector operation
y := alpha*abs(A)*abs(x) + beta*abs(y),
where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix.
This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, 'symbolically' zero components are not perturbed. A zero entry is considered 'symbolic' if all multiplications involved in computing that entry have at least one zero multiplicand.
Parameters
UPLO
          UPLO is INTEGER
           On entry, UPLO specifies whether the upper or lower
           triangular part of the array A is to be referenced as
           follows:
UPLO = BLAS_UPPER Only the upper triangular part of A is to be referenced.
UPLO = BLAS_LOWER Only the lower triangular part of A is to be referenced.
Unchanged on exit.
N
          N is INTEGER
           On entry, N specifies the number of columns of the matrix A.
           N must be at least zero.
           Unchanged on exit.
ALPHA
          ALPHA is REAL .
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.
A
          A is COMPLEX array, dimension ( LDA, n ).
           Before entry, the leading m by n part of the array A must
           contain the matrix of coefficients.
           Unchanged on exit.
LDA
          LDA is INTEGER
           On entry, LDA specifies the first dimension of A as declared
           in the calling (sub) program. LDA must be at least
           max( 1, n ).
           Unchanged on exit.
X
          X is COMPLEX array, dimension
           ( 1 + ( n - 1 )*abs( INCX ) )
           Before entry, the incremented array X must contain the
           vector x.
           Unchanged on exit.
INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
           Unchanged on exit.
BETA
          BETA is REAL .
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
           Unchanged on exit.
Y
          Y is REAL array, dimension
           ( 1 + ( n - 1 )*abs( INCY ) )
           Before entry with BETA non-zero, the incremented array Y
           must contain the vector y. On exit, Y is overwritten by the
           updated vector y.
INCY
          INCY is INTEGER
           On entry, INCY specifies the increment for the elements of
           Y. INCY must not be zero.
           Unchanged on exit.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  Level 2 Blas routine.
-- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. -- Modified for the absolute-value product, April 2006 Jason Riedy, UC Berkeley

real function cla_hercond_c (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension ( * ) C, logical CAPPLY, integer INFO, complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.
Purpose:
 
    CLA_HERCOND_C computes the infinity norm condition number of
    op(A) * inv(diag(C)) where C is a REAL vector.
Parameters
UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
     On entry, the N-by-N matrix A
LDA
          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).
AF
          AF is COMPLEX array, dimension (LDAF,N)
     The block diagonal matrix D and the multipliers used to
     obtain the factor U or L as computed by CHETRF.
LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
     Details of the interchanges and the block structure of D
     as determined by CHETRF.
C
          C is REAL array, dimension (N)
     The vector C in the formula op(A) * inv(diag(C)).
CAPPLY
          CAPPLY is LOGICAL
     If .TRUE. then access the vector C in the formula above.
INFO
          INFO is INTEGER
       = 0:  Successful exit.
     i > 0:  The ith argument is invalid.
WORK
          WORK is COMPLEX array, dimension (2*N).
     Workspace.
RWORK
          RWORK is REAL array, dimension (N).
     Workspace.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

real function cla_hercond_x (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, complex, dimension( * ) X, integer INFO, complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.
Purpose:
 
    CLA_HERCOND_X computes the infinity norm condition number of
    op(A) * diag(X) where X is a COMPLEX vector.
Parameters
UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
A
          A is COMPLEX array, dimension (LDA,N)
     On entry, the N-by-N matrix A.
LDA
          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).
AF
          AF is COMPLEX array, dimension (LDAF,N)
     The block diagonal matrix D and the multipliers used to
     obtain the factor U or L as computed by CHETRF.
LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
     Details of the interchanges and the block structure of D
     as determined by CHETRF.
X
          X is COMPLEX array, dimension (N)
     The vector X in the formula op(A) * diag(X).
INFO
          INFO is INTEGER
       = 0:  Successful exit.
     i > 0:  The ith argument is invalid.
WORK
          WORK is COMPLEX array, dimension (2*N).
     Workspace.
RWORK
          RWORK is REAL array, dimension (N).
     Workspace.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine cla_herfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, real, dimension( * ) C, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, complex, dimension( * ) RES, real, dimension( * ) AYB, complex, dimension( * ) DY, complex, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)

CLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
Purpose:
 
 CLA_HERFSX_EXTENDED improves the computed solution to a system of
 linear equations by performing extra-precise iterative refinement
 and provides error bounds and backward error estimates for the solution.
 This subroutine is called by CHERFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERR_BNDS_NORM
 and ERR_BNDS_COMP for details of the error bounds. Note that this
 subroutine is only responsible for setting the second fields of
 ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters
PREC_TYPE
          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and P
          = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X' or 'E':  Extra
UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
     The number of right-hand-sides, i.e., the number of columns of the
     matrix B.
A
          A is COMPLEX array, dimension (LDA,N)
     On entry, the N-by-N matrix A.
LDA
          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).
AF
          AF is COMPLEX array, dimension (LDAF,N)
     The block diagonal matrix D and the multipliers used to
     obtain the factor U or L as computed by CHETRF.
LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
     Details of the interchanges and the block structure of D
     as determined by CHETRF.
COLEQU
          COLEQU is LOGICAL
     If .TRUE. then column equilibration was done to A before calling
     this routine. This is needed to compute the solution and error
     bounds correctly.
C
          C is REAL array, dimension (N)
     The column scale factors for A. If COLEQU = .FALSE., C
     is not accessed. If C is input, each element of C should be a power
     of the radix to ensure a reliable solution and error estimates.
     Scaling by powers of the radix does not cause rounding errors unless
     the result underflows or overflows. Rounding errors during scaling
     lead to refining with a matrix that is not equivalent to the
     input matrix, producing error estimates that may not be
     reliable.
B
          B is COMPLEX array, dimension (LDB,NRHS)
     The right-hand-side matrix B.
LDB
          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).
Y
          Y is COMPLEX array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by CHETRS.
     On exit, the improved solution matrix Y.
LDY
          LDY is INTEGER
     The leading dimension of the array Y.  LDY >= max(1,N).
BERR_OUT
          BERR_OUT is REAL array, dimension (NRHS)
     On exit, BERR_OUT(j) contains the componentwise relative backward
     error for right-hand-side j from the formula
         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
     where abs(Z) is the componentwise absolute value of the matrix
     or vector Z. This is computed by CLA_LIN_BERR.
N_NORMS
          N_NORMS is INTEGER
     Determines which error bounds to return (see ERR_BNDS_NORM
     and ERR_BNDS_COMP).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.
ERR_BNDS_NORM
          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i))
The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.
ERR_BNDS_COMP
          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i))
The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.
RES
          RES is COMPLEX array, dimension (N)
     Workspace to hold the intermediate residual.
AYB
          AYB is REAL array, dimension (N)
     Workspace.
DY
          DY is COMPLEX array, dimension (N)
     Workspace to hold the intermediate solution.
Y_TAIL
          Y_TAIL is COMPLEX array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.
RCOND
          RCOND is REAL
     Reciprocal scaled condition number.  This is an estimate of the
     reciprocal Skeel condition number of the matrix A after
     equilibration (if done).  If this is less than the machine
     precision (in particular, if it is zero), the matrix is singular
     to working precision.  Note that the error may still be small even
     if this number is very small and the matrix appears ill-
     conditioned.
ITHRESH
          ITHRESH is INTEGER
     The maximum number of residual computations allowed for
     refinement. The default is 10. For 'aggressive' set to 100 to
     permit convergence using approximate factorizations or
     factorizations other than LU. If the factorization uses a
     technique other than Gaussian elimination, the guarantees in
     ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
RTHRESH
          RTHRESH is REAL
     Determines when to stop refinement if the error estimate stops
     decreasing. Refinement will stop when the next solution no longer
     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
     default value is 0.5. For 'aggressive' set to 0.9 to permit
     convergence on extremely ill-conditioned matrices. See LAWN 165
     for more details.
DZ_UB
          DZ_UB is REAL
     Determines when to start considering componentwise convergence.
     Componentwise convergence is only considered after each component
     of the solution Y is stable, which we define as the relative
     change in each component being less than DZ_UB. The default value
     is 0.25, requiring the first bit to be stable. See LAWN 165 for
     more details.
IGNORE_CWISE
          IGNORE_CWISE is LOGICAL
     If .TRUE. then ignore componentwise convergence. Default value
     is .FALSE..
INFO
          INFO is INTEGER
       = 0:  Successful exit.
       < 0:  if INFO = -i, the ith argument to CLA_HERFSX_EXTENDED had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

real function cla_herpvgrw (character*1 UPLO, integer N, integer INFO, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) WORK)

CLA_HERPVGRW
Purpose:
 
 CLA_HERPVGRW computes the reciprocal pivot growth factor
 norm(A)/norm(U). The 'max absolute element' norm is used. If this is
 much less than 1, the stability of the LU factorization of the
 (equilibrated) matrix A could be poor. This also means that the
 solution X, estimated condition numbers, and error bounds could be
 unreliable.
Parameters
UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
INFO
          INFO is INTEGER
     The value of INFO returned from SSYTRF, .i.e., the pivot in
     column INFO is exactly 0.
A
          A is COMPLEX array, dimension (LDA,N)
     On entry, the N-by-N matrix A.
LDA
          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).
AF
          AF is COMPLEX array, dimension (LDAF,N)
     The block diagonal matrix D and the multipliers used to
     obtain the factor U or L as computed by CHETRF.
LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
     Details of the interchanges and the block structure of D
     as determined by CHETRF.
WORK
          WORK is REAL array, dimension (2*N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine clahef (character UPLO, integer N, integer NB, integer KB, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
Purpose:
 
 CLAHEF computes a partial factorization of a complex Hermitian
 matrix A using the Bunch-Kaufman diagonal pivoting method. The
 partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12**H U22**H )
A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**H denotes the conjugate transpose of U.
CLAHEF is an auxiliary routine called by CHETRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NB
          NB is INTEGER
          The maximum number of columns of the matrix A that should be
          factored.  NB should be at least 2 to allow for 2-by-2 pivot
          blocks.
KB
          KB is INTEGER
          The number of columns of A that were actually factored.
          KB is either NB-1 or NB, or N if N <= NB.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, A contains details of the partial factorization.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
If UPLO = 'U': Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L': Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
W
          W is COMPLEX array, dimension (LDW,NB)
LDW
          LDW is INTEGER
          The leading dimension of the array W.  LDW >= max(1,N).
INFO
          INFO is INTEGER
          = 0: successful exit
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley

subroutine clahef_rk (character UPLO, integer N, integer NB, integer KB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
Purpose:
 
 CLAHEF_RK computes a partial factorization of a complex Hermitian
 matrix A using the bounded Bunch-Kaufman (rook) diagonal
 pivoting method. The partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12**H U22**H )
A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L', ( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.
CLAHEF_RK is an auxiliary routine called by CHETRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NB
          NB is INTEGER
          The maximum number of columns of the matrix A that should be
          factored.  NB should be at least 2 to allow for 2-by-2 pivot
          blocks.
KB
          KB is INTEGER
          The number of columns of A that were actually factored.
          KB is either NB-1 or NB, or N if N <= NB.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.
            If UPLO = 'U': the leading N-by-N upper triangular part
            of A contains the upper triangular part of the matrix A,
            and the strictly lower triangular part of A is not
            referenced.
If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
On exit, contains: a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is COMPLEX array, dimension (N)
          On exit, contains the superdiagonal (or subdiagonal)
          elements of the Hermitian block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.
IPIV
          IPIV is INTEGER array, dimension (N)
          IPIV describes the permutation matrix P in the factorization
          of matrix A as follows. The absolute value of IPIV(k)
          represents the index of row and column that were
          interchanged with the k-th row and column. The value of UPLO
          describes the order in which the interchanges were applied.
          Also, the sign of IPIV represents the block structure of
          the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
          diagonal blocks which correspond to 1 or 2 interchanges
          at each factorization step.
If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
W
          W is COMPLEX array, dimension (LDW,NB)
LDW
          LDW is INTEGER
          The leading dimension of the array W.  LDW >= max(1,N).
INFO
          INFO is INTEGER
          = 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine clahef_rook (character UPLO, integer N, integer NB, integer KB, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer LDW, integer INFO)

Purpose:
 
 CLAHEF_ROOK computes a partial factorization of a complex Hermitian
 matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting
 method. The partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12**H U22**H )
A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**H denotes the conjugate transpose of U.
CLAHEF_ROOK is an auxiliary routine called by CHETRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
Parameters
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
NB
          NB is INTEGER
          The maximum number of columns of the matrix A that should be
          factored.  NB should be at least 2 to allow for 2-by-2 pivot
          blocks.
KB
          KB is INTEGER
          The number of columns of A that were actually factored.
          KB is either NB-1 or NB, or N if N <= NB.
A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, A contains details of the partial factorization.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
IPIV
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
If UPLO = 'U': Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L': Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
W
          W is COMPLEX array, dimension (LDW,NB)
LDW
          LDW is INTEGER
          The leading dimension of the array W.  LDW >= max(1,N).
INFO
          INFO is INTEGER
          = 0: successful exit
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
 
  November 2013, Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Author

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