?sbtrd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?sbtrd

Reduces a real symmetric band matrix to tridiagonal form.

Syntax

lapack_int LAPACKE_ssbtrd (int matrix_layout, char vect, char uplo, lapack_int n, lapack_int kd, float* ab, lapack_int ldab, float* d, float* e, float* q, lapack_int ldq);

lapack_int LAPACKE_dsbtrd (int matrix_layout, char vect, char uplo, lapack_int n, lapack_int kd, double* ab, lapack_int ldab, double* d, double* e, double* q, lapack_int ldq);

Include Files

mkl.h

Description

The routine reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A = Q*T*Q^T. The orthogonal matrix Q is determined as a product of Givens rotations.

If required, the routine can also form the matrix Q explicitly.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

vect

Must be 'V', 'N', or 'U'.

If vect = 'V', the routine returns the explicit matrix Q.

If vect = 'N', the routine does not return Q.

If vect = 'U', the routine updates matrix X by forming X*Q.

uplo

Must be 'U' or 'L'.

If uplo = 'U', ab stores the upper triangular part of A.

If uplo = 'L', ab stores the lower triangular part of A.

n

The order of the matrix A (n≥0).

kd

The number of super- or sub-diagonals in A

(kd≥0).

ab, q

ab(size at least max(1, ldab*n) for column major layout and at least max(1, ldab*(kd+ 1)) for row major layout) is an array containing either upper or lower triangular part of the matrix A (as specified by uplo) in band storage format.

q (size max(1, ldq*n)) is an array.

If vect = 'U', the q array must contain an n-by-n matrix X.

If vect = 'N' or 'V', the q parameter need not be set.

ldab

The leading dimension of ab; at least kd+1 for column major layout and n for row major layout .

ldq

The leading dimension of q. Constraints:

ldq≥ max(1, n) if vect = 'V' or 'U';

ldq≥ 1 if vect = 'N'.

Output Parameters

ab

On exit, the diagonal elements of the array ab are overwritten by the diagonal elements of the tridiagonal matrix T. If kd > 0, the elements on the first superdiagonal (if uplo = 'U') or the first subdiagonal (if uplo = 'L') are ovewritten by the off-diagonal elements of T. The rest of ab is overwritten by values generated during the reduction.

d, e, q

Arrays:

d contains the diagonal elements of the matrix T.

The size of d must be at least max(1, n).

e contains the off-diagonal elements of T.

The size of e must be at least max(1, n-1).

q is not referenced if vect = 'N'.

If vect = 'V', q contains the n-by-n matrix Q.

If vect = 'U', q contains the product X* Q.

Return Values

This function returns a value info.

If info=0, the execution is successful.

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

Application Notes

The computed matrix T is exactly similar to a matrix A+E, where ||E||₂ = c(n)*ε*||A||₂, c(n) is a modestly increasing function of n, and ε is the machine precision. The computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that ||E||₂ = O(ε).

The total number of floating-point operations is approximately 6n²*kd if vect = 'N', with 3n³*(kd-1)/kd additional operations if vect = 'V'.

The complex counterpart of this routine is hbtrd.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?sbtrd