Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 7/13/2023
Public

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?trtrs

Solves a system of linear equations with a triangular coefficient matrix, with multiple right-hand sides.

Syntax

call strtrs( uplo, trans, diag, n, nrhs, a, lda, b, ldb, info )

call dtrtrs( uplo, trans, diag, n, nrhs, a, lda, b, ldb, info )

call ctrtrs( uplo, trans, diag, n, nrhs, a, lda, b, ldb, info )

call ztrtrs( uplo, trans, diag, n, nrhs, a, lda, b, ldb, info )

call trtrs( a, b [,uplo] [, trans] [,diag] [,info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine solves for X the following systems of linear equations with a triangular matrix A, with multiple right-hand sides stored in B:

A*X = B

if trans='N',

AT*X = B

if trans='T',

AH*X = B

if trans='C' (for complex matrices only).

Input Parameters

uplo

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

Indicates whether A is upper or lower triangular:

If uplo = 'U', then A is upper triangular.

If uplo = 'L', then A is lower triangular.

trans

CHARACTER*1. Must be 'N' or 'T' or 'C'.

If trans = 'N', then A*X = B is solved for X.

If trans = 'T', then AT*X = B is solved for X.

If trans = 'C', then AH*X = B is solved for X.

diag

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

If diag = 'N', then A is not a unit triangular matrix.

If diag = 'U', then A is unit triangular: diagonal elements of A are assumed to be 1 and not referenced in the array a.

n

INTEGER. The order of A; the number of rows in B; n 0.

nrhs

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

a

REAL for strtrs

DOUBLE PRECISION for dtrtrs

COMPLEX for ctrtrs

DOUBLE COMPLEX for ztrtrs.

The array a(lda,*) contains the matrix A.

The second dimension of a must be at least max(1,n).

b

REAL for strtrs

DOUBLE PRECISION for dtrtrs

COMPLEX for ctrtrs

DOUBLE COMPLEX for ztrtrs.

The array b(ldb,*) contains the matrix B whose columns are the right-hand sides for the systems of equations.

The second dimension of b at least max(1,nrhs).

lda

INTEGER. The leading dimension of a; lda max(1, n).

ldb

INTEGER. The leading 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.

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 trtrs interface are as follows:

a

Stands for argument ap in FORTRAN 77 interface. Holds the matrix 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'.

trans

Must be 'N', 'C', or 'T'. The default value is 'N'.

diag

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

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 |A|

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:


Equation

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 condition number of AT and AH might or might not be equal to κ(A).

The approximate number of floating-point operations for one right-hand side vector b is n2 for real flavors and 4n2 for complex flavors.

To estimate the condition number κ(A), call ?trcon.

To estimate the error in the solution, call ?trrfs.