NAME

complexGTsolve - complex

SYNOPSIS

Functions


subroutine cgtsv (N, NRHS, DL, D, DU, B, LDB, INFO)
 
CGTSV computes the solution to system of linear equations A * X = B for GT matrices subroutine cgtsvx (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
 
CGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Detailed Description

This is the group of complex solve driver functions for GT matrices

Function Documentation

subroutine cgtsv (integer N, integer NRHS, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CGTSV computes the solution to system of linear equations A * X = B for GT matrices
Purpose:
 
 CGTSV  solves the equation
A*X = B,
where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting.
Note that the equation A**T *X = B may be solved by interchanging the order of the arguments DU and DL.
Parameters
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.
DL
          DL is COMPLEX array, dimension (N-1)
          On entry, DL must contain the (n-1) subdiagonal elements of
          A.
          On exit, DL is overwritten by the (n-2) elements of the
          second superdiagonal of the upper triangular matrix U from
          the LU factorization of A, in DL(1), ..., DL(n-2).
D
          D is COMPLEX array, dimension (N)
          On entry, D must contain the diagonal elements of A.
          On exit, D is overwritten by the n diagonal elements of U.
DU
          DU is COMPLEX array, dimension (N-1)
          On entry, DU must contain the (n-1) superdiagonal elements
          of A.
          On exit, DU is overwritten by the (n-1) elements of the first
          superdiagonal of U.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.
          On exit, if INFO = 0, the N-by-NRHS 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
          > 0:  if INFO = i, U(i,i) is exactly zero, and the solution
                has not been computed.  The factorization has not been
                completed unless i = N.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine cgtsvx (character FACT, character TRANS, integer N, integer NRHS, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( * ) DLF, complex, dimension( * ) DF, complex, dimension( * ) DUF, complex, dimension( * ) DU2, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CGTSVX computes the solution to system of linear equations A * X = B for GT matrices
Purpose:
 
 CGTSVX uses the LU factorization to compute the solution to a complex
 system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
 where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
 matrices.
Error bounds on the solution and a condition estimate are also provided.
Description:
 
 The following steps are performed:
1. If FACT = 'N', the LU decomposition is used to factor the matrix A as A = L * U, where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
3. The system of equations is solved for X using the factored form of A.
4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
Parameters
FACT
          FACT is CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored form
                  of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
                  be modified.
          = 'N':  The matrix will be copied to DLF, DF, and DUF
                  and factored.
TRANS
          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)
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.
DL
          DL is COMPLEX array, dimension (N-1)
          The (n-1) subdiagonal elements of A.
D
          D is COMPLEX array, dimension (N)
          The n diagonal elements of A.
DU
          DU is COMPLEX array, dimension (N-1)
          The (n-1) superdiagonal elements of A.
DLF
          DLF is COMPLEX array, dimension (N-1)
          If FACT = 'F', then DLF is an input argument and on entry
          contains the (n-1) multipliers that define the matrix L from
          the LU factorization of A as computed by CGTTRF.
If FACT = 'N', then DLF is an output argument and on exit contains the (n-1) multipliers that define the matrix L from the LU factorization of A.
DF
          DF is COMPLEX array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.
If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
DUF
          DUF is COMPLEX array, dimension (N-1)
          If FACT = 'F', then DUF is an input argument and on entry
          contains the (n-1) elements of the first superdiagonal of U.
If FACT = 'N', then DUF is an output argument and on exit contains the (n-1) elements of the first superdiagonal of U.
DU2
          DU2 is COMPLEX array, dimension (N-2)
          If FACT = 'F', then DU2 is an input argument and on entry
          contains the (n-2) elements of the second superdiagonal of
          U.
If FACT = 'N', then DU2 is an output argument and on exit contains the (n-2) elements of the second superdiagonal of U.
IPIV
          IPIV is INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the LU factorization of A as
          computed by CGTTRF.
If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the LU factorization of A; row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required.
B
          B is COMPLEX array, dimension (LDB,NRHS)
          The N-by-NRHS 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)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
RCOND
          RCOND is REAL
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working
          precision.  This condition is indicated by a return code of
          INFO > 0.
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
          > 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has not been completed unless i = N, but the
                       factor U is exactly singular, so the solution
                       and error bounds could not be computed.
                       RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the
                       value of RCOND would suggest.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Author

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