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',

AT*X = B

if trans='T',

AH*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 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.

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, ldb*nrhs) for column major layout and max(1, n*ldb) 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 ldbnrhs 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:


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 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.

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