?gttrs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?gttrs

Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by ?gttrf.

Syntax

lapack_int LAPACKE_sgttrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const float * dl , const float * d , const float * du , const float * du2 , const lapack_int * ipiv , float * b , lapack_int ldb );

lapack_int LAPACKE_dgttrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const double * dl , const double * d , const double * du , const double * du2 , const lapack_int * ipiv , double * b , lapack_int ldb );

lapack_int LAPACKE_cgttrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const lapack_complex_float * dl , const lapack_complex_float * d , const lapack_complex_float * du , const lapack_complex_float * du2 , const lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zgttrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const lapack_complex_double * dl , const lapack_complex_double * d , const lapack_complex_double * du , const lapack_complex_double * du2 , const lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );

Include Files

mkl.h

Description

The routine solves for X the following systems of linear equations with multiple right hand sides:

A*X = B	if `trans`='N',
`A^T*X = B`	if `trans`='T',
A^H`*X` = B	if `trans`='C' (for complex matrices only).

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

Input Parameters

matrix_layout	Specifies whether matrix storage layout for array b is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
trans	Must be 'N' or 'T' or 'C'. Indicates the form of the equations: If `trans = 'N'`, then `AX = B` is solved for `X`. If `trans = 'T'`, then `A`^T`X` = `B` is solved for `X`. If `trans = 'C'`, then `A`^H`*X` = `B` is solved for `X`.
n	The order of `A`; `n`≥ 0.
nrhs	The number of right-hand sides, that is, the number of columns in `B; nrhs≥ 0`.
dl,d,du,du2	Arrays: `dl(n -1)`, `d(n)`, `du(n -1)`, `du2(n -2)`. The array `dl` contains the `(n - 1)` multipliers that define the matrix `L` from the `LU` factorization of `A`. The array `d` contains the `n` diagonal elements of the upper triangular matrix `U` from the `LU` factorization of `A`. The array `du` contains the (`n` - 1) elements of the first superdiagonal of `U`. The array `du2` contains the (`n` - 2) elements of the second superdiagonal of `U`.
b	Array of size max(1, ldbnrhs) for column major layout and max(1, nldb) for row major layout. Contains the matrix `B` whose columns are the right-hand sides for the systems of equations.
ldb	The leading dimension of `b;` `ldb≥ max(1, n)` for column major layout and `ldb≥nrhs` for row major layout.
ipiv	Array, size (`n`). The `ipiv` array, as returned by ?gttrf.

Output Parameters

b	Overwritten by the solution matrix `X`.

Return Values

This function returns a value info.

If info=0, the execution is successful.

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

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|L||U|

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 condition number of A^T and A^H might or might not be equal to κ_∞(A).

The approximate number of floating-point operations for one right-hand side vector b is 7n (including n divisions) for real flavors and 34n (including 2n divisions) for complex flavors.

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

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

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

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