?pbtrs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?pbtrs

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

Syntax

call spbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call dpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call cpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call zpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call pbtrs( ab, b [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

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

`A = U^TU` for real data, `A = U^HU` for complex data	if `uplo='U'`
`A = LL^T` for real data, `A = LL^H` 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 ?pbtrf to compute the Cholesky factorization of A in the band storage form.

Input Parameters

uplo	CHARACTER1. Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: If `uplo = 'U'`, `U` is stored in `ab`, where `A` = `U`^T`U` for real matrices and `A` = `U`^H`U` for complex matrices. If `uplo = 'L'`, `L` is stored in `ab`, where `A` = `L``L`^T for real matrices and `A` = `L`*`L`^H for complex matrices.
n	INTEGER. The order of matrix `A`; `n`≥ 0.
kd	INTEGER. The number of superdiagonals or subdiagonals in the matrix `A`; `kd`≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs`≥ 0.
ab, b	REAL for spbtrs DOUBLE PRECISION for dpbtrs COMPLEX for cpbtrs DOUBLE COMPLEX for zpbtrs. Arrays: `ab(ldab,)`, `b(ldb,)`. The array `ab` contains the Cholesky factor, as returned by the factorization routine, in band storage form. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The second dimension of `ab` must be at least `max(1, n)`, and the second dimension of `b` at least `max(1,nrhs)`.
ldab	INTEGER. The leading dimension of the array `ab`; `ldab`≥`kd` +1.
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 pbtrs interface are as follows:

ab	Holds the array `A` of size `(kd+1,n)`.
b	Holds the matrix `B` of size (`n`, `nrhs`).
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(kd + 1)ε P|U^H||U| or |E| ≤ c(kd + 1)ε P|L^H||L|

c(k) is a modest linear function of k, 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 approximate number of floating-point operations for one right-hand side vector is 4n*kd for real flavors and 16n*kd for complex flavors.

To estimate the condition number κ_∞(A), call ?pbcon.

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

Parent topic: Solving Systems of Linear Equations: LAPACK Computational Routines

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