?sbtrd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?sbtrd

Reduces a real symmetric band matrix to tridiagonal form.

Syntax

call ssbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)

call dsbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)

call sbtrd(ab[, q] [,vect] [,uplo] [,info])

Include Files

mkl.fi, lapack.f90

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

vect

CHARACTER*1. 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

CHARACTER*1. 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

INTEGER. The order of the matrix A (n≥0).

kd

INTEGER. The number of super- or sub-diagonals in A

(kd≥0).

ab, q, work

REAL for ssbtrd

DOUBLE PRECISION for dsbtrd.

ab(ldab,*) is an array containing either upper or lower triangular part of the matrix A (as specified by uplo) in band storage format.

The second dimension of ab must be at least max(1, n).

q(ldq,*) 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.

The second dimension of q must be at least max(1, n).

work(*) is a workspace array.

The dimension of work must be at least max(1, n).

ldab

INTEGER. The leading dimension of ab; at least kd+1 .

ldq

INTEGER. 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

REAL for ssbtrd

DOUBLE PRECISION for dsbtrd.

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(ldq,*) 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.

The second dimension of q must be:

at least max(1, n) if vect = 'V';

at least 1 if vect = 'N'.

info

INTEGER.

If info = 0, the execution is successful.

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

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine sbtrd interface are the following:

ab

Holds the array A of size (kd+1,n).

q

Holds the matrix Q of size (n,n).

uplo

Must be 'U' or 'L'. The default value is 'U'.

vect

If omitted, this argument is restored based on the presence of argument q as follows: vect = 'V', if q is present, vect = 'N', if q is omitted.

If present, vect must be equal to 'V' or 'U' and the argument q must also be present. Note that there will be an error condition if vect is present and q omitted.

Note that diagonal (d) and off-diagonal (e) elements of the matrix T are omitted because they are kept in the matrix A on exit.

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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?sbtrd