?dtsvb

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?dtsvb

Computes the solution to the system of linear equations with a diagonally dominant tridiagonal coefficient matrix A and multiple right-hand sides.

Syntax

call sdtsvb( n, nrhs, dl, d, du, b, ldb, info )

call ddtsvb( n, nrhs, dl, d, du, b, ldb, info )

call cdtsvb( n, nrhs, dl, d, du, b, ldb, info )

call zdtsvb( n, nrhs, dl, d, du, b, ldb, info )

call dtsvb( dl, d, du, b [, info])

Include Files

mkl.fi, lapack.f90

Description

The ?dtsvb routine solves a system of linear equations A*X = B for X, where A is an n-by-n diagonally dominant tridiagonal matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions. The routine uses the BABE (Burning At Both Ends) algorithm.

Note that the equation A^T*X = B may be solved by interchanging the order of the arguments du and dl.

Input Parameters

n	INTEGER. The order of `A`, the number of rows in `B`; n≥ 0.
nrhs	INTEGER. The number of right-hand sides, the number of columns in `B`; nrhs≥ 0.
dl, d, du, b	REAL for sdtsvb DOUBLE PRECISION for ddtsvb COMPLEX for cdtsvb DOUBLE COMPLEX for zdtsvb. Arrays: dl (size n - 1), d (size n), du (size n - 1), b(size ldb,*). The array dl contains the `(n - 1)` subdiagonal elements of `A`. The array d contains the diagonal elements of `A`. The array du contains the `(n - 1)` superdiagonal elements of `A`. The array b contains the matrix `B` whose columns are the right-hand sides for the systems 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

dl	Overwritten by the `(n-1)` elements of the subdiagonal of the lower triangular matrices `L`₁, `L`₂ from the factorization of `A` (see dttrfb).
d	Overwritten by the n diagonal element reciprocals of `U`.
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. If `info = i`, `u_ii` is exactly zero, and the solution has not been computed. The factorization has not been completed unless `i = n`.

Application Notes

A diagonally dominant tridiagonal system is defined such that |d_i| > |dl_i-1| + |du_i| for any i:

1 < i < n, and |d₁| > |du₁|, |d_n| > |dl_n-1|

The underlying BABE algorithm is designed for diagonally dominant systems. Such systems have no numerical stability issue unlike the canonical systems that use elimination with partial pivoting (see ?gtsv). The diagonally dominant systems are much faster than the canonical systems.

NOTE:

The current implementation of BABE has a potential accuracy issue on very small or large data close to the underflow or overflow threshold respectively. Scale the matrix before applying the solver in the case of such input data.
Applying the ?dtsvb factorization to non-diagonally dominant systems may lead to an accuracy loss, or false singularity detected due to no pivoting.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?dtsvb