---
name: "cptsvx"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/cptsvx/3lapack"
---

# cptsvx(3lapack)

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 = 'N', 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. = 'F':  On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified. = 'N':  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 = 'F', 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 = 'N', 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 = 'F', 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 = 'N', 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.