NAME
complexOTHEReigen - complex Other Eigenvalue routinesSYNOPSIS
Functions
subroutine cggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)
Detailed Description
This is the group of complex Other Eigenvalue routinesFunction Documentation
subroutine cggglm (integer N, integer M, integer P, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) D, complex, dimension( * ) X, complex, dimension( * ) Y, complex, dimension( * ) WORK, integer LWORK, integer INFO)
CGGGLM Purpose:CGGGLM solves a general Gauss-Markov linear model (GLM) problem: minimize || y ||_2 subject to d = A*x + B*y x where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. It is assumed that M <= N <= M+P, and rank(A) = M and rank( A B ) = N. Under these assumptions, the constrained equation is always consistent, and there is a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices (A, B) given by A = Q*(R), B = Q*T*Z. (0) In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem minimize || inv(B)*(d-A*x) ||_2 x where inv(B) denotes the inverse of B.
N
Author
N is INTEGER The number of rows of the matrices A and B. N >= 0.M
M is INTEGER The number of columns of the matrix A. 0 <= M <= N.P
P is INTEGER The number of columns of the matrix B. P >= N-M.A
A is COMPLEX array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the upper triangular part of the array A contains the M-by-M upper triangular matrix R.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).B
B is COMPLEX array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)th subdiagonal contain the N-by-P upper trapezoidal matrix T.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).D
D is COMPLEX array, dimension (N) On entry, D is the left hand side of the GLM equation. On exit, D is destroyed.X
X is COMPLEX array, dimension (M)Y
Y is COMPLEX array, dimension (P) On exit, X and Y are the solutions of the GLM problem.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 >= max(1,N+M+P). For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB is an upper bound for the optimal blocksizes for CGEQRF, CGERQF, CUNMQR and CUNMRQ. 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. = 1: the upper triangular factor R associated with A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M; the least squares solution could not be computed. = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank( A B ) < N; the least squares solution could not be computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chbev (character JOBZ, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)
CHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHBEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.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 matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, 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). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (N)RWORK
RWORK is REAL array, dimension (max(1,3*N-2))INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chbev_2stage (character JOBZ, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)
CHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. 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.KD
KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, 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). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension LWORK On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (max(1,3*N-2))INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
All details about the 2stage techniques are available in: 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 chbevd (character JOBZ, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
CHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHBEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.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 matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, 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). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).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. If N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least N. If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.LRWORK
LRWORK is INTEGER The dimension of array RWORK. If N <= 1, LRWORK must be at least 1. If JOBZ = 'N' and N > 1, LRWORK must be at least N. If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORK
LIWORK is INTEGER The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N . If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK 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, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
CHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. 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.KD
KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, 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). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).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 the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.LRWORK
LRWORK is INTEGER The dimension of array RWORK. If N <= 1, LRWORK must be at least 1. If JOBZ = 'N' and N > 1, LRWORK must be at least N. If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORK
LIWORK is INTEGER The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N . If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK 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, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
All details about the 2stage techniques are available in: 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 chbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGE
RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.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 matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, 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). On exit, AB is overwritten by values generated during the reduction to tridiagonal form.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.Q
Q is COMPLEX array, dimension (LDQ, N) If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.LDQ
LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).VL
VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AB to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.M
M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.W
W is REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, max(1,M)) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (N)RWORK
RWORK is REAL array, dimension (7*N)IWORK
IWORK is INTEGER array, dimension (5*N)IFAIL
IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.RANGE
RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.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 matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, 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). On exit, AB is overwritten by values generated during the reduction to tridiagonal form.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.Q
Q is COMPLEX array, dimension (LDQ, N) If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.LDQ
LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).VL
VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AB to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.M
M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.W
W is REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, max(1,M)) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (LWORK)LWORK
LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (7*N)IWORK
IWORK is INTEGER array, dimension (5*N)IFAIL
IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.
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Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
All details about the 2stage techniques are available in: 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 chbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)
CHBGV Purpose:CHBGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.N
N is INTEGER The order of the matrices A and B. N >= 0.KA
KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.KB
KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.BB
BB is COMPLEX array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.LDBB
LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.WORK
WORK is COMPLEX array, dimension (N)RWORK
RWORK is REAL array, dimension (3*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is: <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
CHBGVD Purpose:CHBGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.N
N is INTEGER The order of the matrices A and B. N >= 0.KA
KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.KB
KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.BB
BB is COMPLEX array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.LDBB
LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.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. If N <= 1, LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= N. If JOBZ = 'V' and N > 1, LWORK >= 2*N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (MAX(1,LRWORK)) On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.LRWORK
LRWORK is INTEGER The dimension of array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = 'N' and N > 1, LRWORK >= N. If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.LIWORK
LIWORK is INTEGER The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK 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, and i is: <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ.
of Kentucky, USA
subroutine chbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA, integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB, complex, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
CHBGVX Purpose:CHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGE
RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.N
N is INTEGER The order of the matrices A and B. N >= 0.KA
KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.KB
KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.AB
AB is COMPLEX array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.LDAB
LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.BB
BB is COMPLEX array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.LDBB
LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.Q
Q is COMPLEX array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.LDQ
LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).VL
VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').M
M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.WORK
WORK is COMPLEX array, dimension (N)RWORK
RWORK is REAL array, dimension (7*N)IWORK
IWORK is INTEGER array, dimension (5*N)IFAIL
IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is: <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ.
of Kentucky, USA
subroutine chpev (character JOBZ, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)
CHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHPEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.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.AP
AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (max(1, 2*N-1))RWORK
RWORK is REAL array, dimension (max(1, 3*N-2))INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chpevd (character JOBZ, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
CHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHPEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.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.AP
AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the required LWORK.LWORK
LWORK is INTEGER The dimension of array WORK. If N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least N. If JOBZ = 'V' and N > 1, LWORK must be at least 2*N. If LWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the required LRWORK.LRWORK
LRWORK is INTEGER The dimension of array RWORK. If N <= 1, LRWORK must be at least 1. If JOBZ = 'N' and N > 1, LRWORK must be at least N. If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the required LIWORK.LIWORK
LIWORK is INTEGER The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK 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, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chpevx (character JOBZ, character RANGE, character UPLO, integer N, complex, dimension( * ) AP, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices Purpose:CHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
JOBZ
Author
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGE
RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.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.AP
AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.VL
VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.M
M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.W
W is REAL array, dimension (N) If INFO = 0, the selected eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, max(1,M)) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (2*N)RWORK
RWORK is REAL array, dimension (7*N)IWORK
IWORK is INTEGER array, dimension (5*N)IFAIL
IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chpgv (integer ITYPE, character JOBZ, character UPLO, integer N, complex, dimension( * ) AP, complex, dimension( * ) BP, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)
CHPGV Purpose:CHPGV computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite.
ITYPE
Author
ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*xJOBZ
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.N
N is INTEGER The order of the matrices A and B. N >= 0.AP
AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.BP
BP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (max(1, 2*N-1))RWORK
RWORK is REAL array, dimension (max(1, 3*N-2))INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: CPPTRF or CHPEV returned an error code: <= N: if INFO = i, CHPEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not convergeto zero; > N: if INFO = N + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine chpgvd (integer ITYPE, character JOBZ, character UPLO, integer N, complex, dimension( * ) AP, complex, dimension( * ) BP, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
CHPGVD Purpose:CHPGVD computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
ITYPE
Author
ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*xJOBZ
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.N
N is INTEGER The order of the matrices A and B. N >= 0.AP
AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.BP
BP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.W
W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the required LWORK.LWORK
LWORK is INTEGER The dimension of array WORK. If N <= 1, LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= N. If JOBZ = 'V' and N > 1, LWORK >= 2*N. If LWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.RWORK
RWORK is REAL array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the required LRWORK.LRWORK
LRWORK is INTEGER The dimension of array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = 'N' and N > 1, LRWORK >= N. If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the required LIWORK.LIWORK
LIWORK is INTEGER The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: CPPTRF or CHPEVD returned an error code: <= N: if INFO = i, CHPEVD failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not convergeto zero; > N: if INFO = N + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ.
of Kentucky, USA
subroutine chpgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, complex, dimension( * ) AP, complex, dimension( * ) BP, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
CHPGVX Purpose:CHPGVX computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
ITYPE
Author
ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*xJOBZ
JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGE
RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.UPLO
UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.N
N is INTEGER The order of the matrices A and B. N >= 0.AP
AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.BP
BP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.VL
VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').M
M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.W
W is REAL array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.Z
Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'N', then Z is not referenced. If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORK
WORK is COMPLEX array, dimension (2*N)RWORK
RWORK is REAL array, dimension (7*N)IWORK
IWORK is INTEGER array, dimension (5*N)IFAIL
IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: CPPTRF or CHPEVX returned an error code: <= N: if INFO = i, CHPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ.
of Kentucky, USA
Author
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