# ?sptrs

Solves a system of linear equations with a UDU- or LDL-factored symmetric matrix using packed storage.

## Syntax

FORTRAN 77:

call ssptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call dsptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call csptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call zsptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

Fortran 95:

call sptrs( a, b, ipiv [, uplo] [,info] )

## Description

This routine solves for X the system of linear equations `A*X = B` with a symmetric matrix A, given the Bunch-Kaufman factorization of A:

if `uplo='U'`,

  A = `PUDUTPT`

if `uplo='L'`,

  A = `PLDLTPT`,

where P is a permutation matrix, U and L are upper and lower packed triangular matrices with unit diagonal, and D is a symmetric block-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix B. You must supply the factor U (or L) and the array ipiv returned by the factorization routine ?sptrf.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates how the input matrix A has been factored:

  If `uplo = 'U'`, the array ap stores the packed factor U of the factorization A = `P*U*D*UT*PT`. If `uplo = 'L'`, the array ap stores the packed factor L of the factorization A = `P*L*D*LT*PT`.

n

  INTEGER. The order of matrix A; n ≥ 0.

nrhs

  INTEGER. The number of right-hand sides; nrhs ≥ 0.

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`. The ipiv array, as returned by ?sptrf.

ap, b

  REAL for ssptrs

  DOUBLE PRECISION for dsptrs

  COMPLEX for csptrs

  DOUBLE COMPLEX for zsptrs.

  Arrays: `ap(*)`, `b(ldb,*)`.

  The dimension of ap must be at least max(1,n(n+1)/2). The array ap contains the factor U or L, as specified by uplo, in packed storage (see Matrix Storage Schemes).

  The array b contains the matrix B whose columns are the right-hand sides for the system of equations. The second dimension of b must be at least `max(1, nrhs)`.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

## Output Parameters

b

  Overwritten by the solution matrix X.

info

  INTEGER. If `info=0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their Fortran 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine sptrs interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size `(n*(n+1)/2)`.

b

  Holds the matrix B of size (n, nrhs).

ipiv

  Holds the vector of length (n).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations `(A + E)x = b`, where
`|E| ≤ c(n)ε P|U||D||UT|PT or |E| ≤ c(n)ε P|L||D||LT|PT`

`c(n)` is a modest linear function of n, and ε is the machine precision.

If `x0` is the true solution, the computed solution x satisfies this error bound:

where `cond(A,x)`= || |A-1||A| |x| ||∞ / ||x||∞ ≤ ||A-1||∞ ||A||∞ = κ∞(A).

Note that `cond(A,x)` can be much smaller than `κ∞(A)`.

The total number of floating-point operations for one right-hand side vector is approximately `2n2` for real flavors or `8n2` for complex flavors.

To estimate the condition number `κ∞(A)`, call ?spcon.

To refine the solution and estimate the error, call ?sprfs.

Parent topic: Routines for Solving Systems of Linear Equations