?dttrfb

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?dttrfb

Computes the factorization of a diagonally dominant tridiagonal matrix.

Syntax

void sdttrfb (const MKL_INT * n , float * dl , float * d , const float * du , MKL_INT * info );

void ddttrfb (const MKL_INT * n , double * dl , double * d , const double * du , MKL_INT * info );

void cdttrfb (const MKL_INT * n , MKL_Complex8 * dl , MKL_Complex8 * d , const MKL_Complex8 * du , MKL_INT * info );

void zdttrfb_ (const MKL_INT * n , MKL_Complex16 * dl , MKL_Complex16 * d , const MKL_Complex16 * du , MKL_INT * info );

Include Files

mkl.h

Description

The ?dttrfb routine computes the factorization of a real or complex tridiagonal matrix A with the BABE (Burning At Both Ends) algorithm without pivoting. The factorization has the form

A = L₁*U*L₂

where

L₁ and L₂ are unit lower bidiagonal with k and n - k - 1 subdiagonal elements, respectively, where k = n/2, and
U is an upper bidiagonal matrix with nonzeroes in only the main diagonal and first superdiagonal.

Input Parameters

The order of the matrix A; n≥ 0.

dl, d, du

Arrays containing elements of A.

The array dl of dimension (n - 1) contains the subdiagonal elements of A.

The array d of dimension n contains the diagonal elements of A.

The array du of dimension (n - 1) contains the superdiagonal elements of A.

Output Parameters

dl	Overwritten by the `(n -1)` multipliers that define the matrix `L` from the `LU` factorization of `A`.
d	Overwritten by the n diagonal element reciprocals of the upper triangular matrix `U` from the factorization of `A`.
du	Overwritten by the `(n-1)` elements of the superdiagonal of `U`.
info	If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i, u_ii` is 0. The factorization has been completed, but `U` is exactly singular. Division by zero will occur if you use the factor `U` for solving a system of linear equations.

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 are free from the numerical stability issue unlike the canonical systems that use elimination with partial pivoting (see ?gttrf). 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 ?dttrfb factorization to non-diagonally dominant systems may lead to an accuracy loss, or false singularity detected due to no pivoting.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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