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

# cspsvx(3lapack)

uses the diagonal pivoting factorization A = U\*D\*U\*\*T or A = L\*D\*L\*\*T to compute the solution to a complex system of linear equations A \* X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices

## NAME

CSPSVX - uses the diagonal pivoting factorization A = U\*D\*U\*\*T or A = L\*D\*L\*\*T to compute the solution to a complex system of linear equations A \* X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices

## SYNOPSIS

- SUBROUTINE CSPSVX(

  FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

- CHARACTER

  FACT, UPLO

- INTEGER

  INFO, LDB, LDX, N, NRHS

- REAL

  RCOND

- INTEGER

  IPIV( \* )

- REAL

  BERR( \* ), FERR( \* ), RWORK( \* )

- COMPLEX

  AFP( \* ), AP( \* ), B( LDB, \* ), WORK( \* ), X( LDX, \* )

## PURPOSE

CSPSVX uses the diagonal pivoting factorization A = U\*D\*U\*\*T or A = L\*D\*L\*\*T to compute the solution to a complex system of linear equations A \* X = B, where A is an N-by-N symmetric matrix stored in packed format 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 diagonal pivoting method is used to factor A as\
       A = U \* D \* U\*\*T,  if UPLO = 'U', or\
\
       A = L \* D \* L\*\*T,  if UPLO = 'L',\
\
    where U (or L) is a product of permutation and unit upper (lower)\
    triangular matrices and D is symmetric and block diagonal with\
    1-by-1 and 2-by-2 diagonal blocks.\
 2\. If some D(i,i)=0, so that D 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.\

## ARGUMENTS

- FACT (input) CHARACTER\*1

  Specifies whether or not the factored form of A has been supplied on entry. = 'F':  On entry, AFP and IPIV contain the factored form of A.  AP, AFP and IPIV will not be modified. = 'N':  The matrix A will be copied to AFP and factored.

- UPLO (input) CHARACTER\*1

  \= 'U':  Upper triangle of A is stored;\
   \= 'L':  Lower triangle of A is stored.

- N (input) INTEGER

  The number of linear equations, i.e., 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.

- AP (input) COMPLEX array, dimension (N\*(N+1)/2)

  The upper or lower triangle of the symmetric 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. See below for further details.

- AFP (input or output) COMPLEX array, dimension (N\*(N+1)/2)

  If FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U\*D\*U\*\*T or A = L\*D\*L\*\*T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A. If FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U\*D\*U\*\*T or A = L\*D\*L\*\*T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A.

- IPIV (input or output) INTEGER array, dimension (N)

  If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CSPTRF. If IPIV(k) \> 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) \< 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) \< 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CSPTRF.

- 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 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 (output) 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 (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 (2\*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:  D(i,i) is exactly zero.  The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+1: D 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.

## FURTHER DETAILS

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':\
 Two-dimensional storage of the symmetric matrix A:\
\
    a11 a12 a13 a14\
\
        a22 a23 a24\
\
            a33 a34     (aij = aji)\
\
                a44\
 Packed storage of the upper triangle of A:\
 AP = \[ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 \]