cptsvx (l) - Linux Manuals

cptsvx: uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices

NAME

CPTSVX - uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices

SYNOPSIS

SUBROUTINE CPTSVX(
FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

    
CHARACTER FACT

    
INTEGER INFO, LDB, LDX, N, NRHS

    
REAL RCOND

    
REAL BERR( * ), D( * ), DF( * ), FERR( * ), RWORK( * )

    
COMPLEX B( LDB, * ), E( * ), EF( * ), WORK( * ), X( LDX, * )

PURPOSE

CPTSVX uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix 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 = aqNaq, the matrix A is factored as A = L*D*L**H, where L
is a unit lower bidiagonal matrix and D is diagonal.  The
factorization can also be regarded as having the form

A = U**H*D*U.
2. If the leading i-by-i principal minor is not positive definite,
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.

ARGUMENTS

FACT (input) CHARACTER*1
Specifies whether or not the factored form of the matrix A is supplied on entry. = aqFaq: On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified. = aqNaq: The matrix A will be copied to DF and EF and factored.
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
D (input) REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.
E (input) COMPLEX array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.
DF (input or output) REAL array, dimension (N)
If FACT = aqFaq, then DF is an input argument and on entry contains the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A. If FACT = aqNaq, then DF is an output argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A.
EF (input or output) COMPLEX array, dimension (N-1)
If FACT = aqFaq, then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. If FACT = aqNaq, then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A.
B (input) COMPLEX array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) REAL
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 (output) REAL array, dimension (NRHS)
The 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).
BERR (output) 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 (workspace) COMPLEX array, dimension (N)
RWORK (workspace) REAL array, dimension (N)
INFO (output) 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 leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been 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.

Pages related to cptsvx

  • cptsvx (3)
  • cptsv (l) - computes the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices
  • cptcon (l) - computes the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite tridiagonal matrix using the factorization A = L*D*L**H or A = U**H*D*U computed by CPTTRF
  • cpteqr (l) - computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor
  • cptrfs (l) - improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution
  • cpttrf (l) - computes the L*D*Laq factorization of a complex Hermitian positive definite tridiagonal matrix A
  • cpttrs (l) - solves a tridiagonal system of the form A * X = B using the factorization A = Uaq*D*U or A = L*D*Laq computed by CPTTRF
  • cptts2 (l) - solves a tridiagonal system of the form A * X = B using the factorization A = Uaq*D*U or A = L*D*Laq computed by CPTTRF