?sptrs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?sptrs

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

Syntax

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 )

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

Include Files

mkl.fi, lapack.f90

Description

The 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` = `UDU^T`
if `uplo='L'`,	`A` = `LDL^T`,

where 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	CHARACTER1. 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` = `UDU^T`. If `uplo = 'L'`, the array `ap` stores the packed factor `L` of the factorization `A` = `LD*L^T`.
n	INTEGER. The order of matrix `A`; `n`≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs`≥ 0.
ipiv	INTEGER. Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?sptrf.
ap	REAL for ssptrs DOUBLE PRECISION for dsptrs COMPLEX for csptrs DOUBLE COMPLEX for zsptrs. The dimension of array `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).
b	REAL for ssptrs DOUBLE PRECISION for dsptrs COMPLEX for csptrs DOUBLE COMPLEX for zsptrs. The array `b(ldb,*)` 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 leading dimension of `b`; `ldb≥ max(1, n)`.

Output Parameters

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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine sptrs interface are as follows:

ap	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||U^T|P^T or |E| ≤ c(n)ε P|L||D||L^T|P^T

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: Solving Systems of Linear Equations: LAPACK Computational Routines

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