# ?pptrs

Solves a system of linear equations with a packed Cholesky-factored symmetric (Hermitian) positive-definite matrix.

## Syntax

FORTRAN 77:

call spptrs( uplo, n, nrhs, ap, b, ldb, info )

call dpptrs( uplo, n, nrhs, ap, b, ldb, info )

call cpptrs( uplo, n, nrhs, ap, b, ldb, info )

call zpptrs( uplo, n, nrhs, ap, b, ldb, info )

Fortran 95:

call pptrs( a, b [,uplo] [,info] )

## Description

This routine solves for X the system of linear equations `A*X = B` with a packed symmetric positive-definite or, for complex data, Hermitian positive-definite matrix A, given the Cholesky factorization of A:

`A = UT*U` for real data, `A = UH*U` for complex data

  If uplo = 'U'

`A = L*LT` for real data, `A = L*LH` for complex data

  If uplo = 'L',

where L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix B.

Before calling this routine, you must call ?pptrf to compute the Cholesky factorization of A.

## Input Parameters

uplo

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

  Indicates how the input matrix A has been factored:

  If `uplo = 'U'`, the upper triangle of A is stored.

  If `uplo = 'L'`, the lower triangle of A is stored.

n

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

nrhs

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

ap, b

  REAL for spptrs

  DOUBLE PRECISION for dpptrs

  COMPLEX for cpptrs

  DOUBLE COMPLEX for zpptrs.

  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 systems 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 pptrs 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).

uplo

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

## Application Notes

If `uplo = 'U'`, the computed solution for each right-hand side b is the exact solution of a perturbed system of equations `(A + E)x = b`, where
`|E| ≤ c(n)ε |UH||U|`

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

A similar estimate holds for `uplo = 'L'`.

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 approximate number of floating-point operations for one right-hand side vector b is `2n2` for real flavors and `8n2` for complex flavors.

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

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

Parent topic: Routines for Solving Systems of Linear Equations