?sptrd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?sptrd

Reduces a real symmetric matrix to tridiagonal form using packed storage.

Syntax

call ssptrd(uplo, n, ap, d, e, tau, info)

call dsptrd(uplo, n, ap, d, e, tau, info)

call sptrd(ap, tau [,uplo] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine reduces a packed real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A = Q*T*Q^T. The orthogonal matrix Q is not formed explicitly but is represented as a product of n-1 elementary reflectors. Routines are provided for working with Q in this representation. See Application Notes below for details.

Input Parameters

uplo

CHARACTER*1. Must be 'U' or 'L'.

If uplo = 'U', ap stores the packed upper triangle of A.

If uplo = 'L', ap stores the packed lower triangle of A.

n

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

ap

REAL for ssptrd

DOUBLE PRECISION for dsptrd.

Array, size at least max(1, n(n+1)/2). Contains either upper or lower triangle of A (as specified by uplo) in the packed form described in Matrix Storage Schemes.

Output Parameters

ap

Overwritten by the tridiagonal matrix T and details of the orthogonal matrix Q, as specified by uplo.

d, e, tau

REAL for ssptrd

DOUBLE PRECISION for dsptrd.

Arrays:

d(*) contains the diagonal elements of the matrix T.

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

e(*) contains the off-diagonal elements of T.

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

tau(*) Stores (n-1) scalars that define elementary reflectors in decomposition of the matrix Q in a product of n-1 reflectors.

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

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 sptrd interface are the following:

ap: Holds the array A of size (n*(n+1)/2).
tau: Holds the vector with the number of elements n-1.
uplo: Must be 'U' or 'L'. The default value is 'U'.

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 matrix Q is represented as a product of n-1 elementary reflectors, as follows :

If uplo = 'U', Q = H(n-1) ... H(2)H(1)

Each H(i) has the form

H(i) = I - tau*v*v^T

where tau is a real scalar and v is a real vector with v(i+1:n) = 0 and v(i) = 1.

On exit, tau is stored in tau(i), and v(1:i-1) is stored in AP, overwriting A(1:i-1, i+1).
If uplo = 'L', Q = H(1)H(2) ... H(n-1)

Each H(i) has the form

H(i) = I - tau*v*v^T

where tau is a real scalar and v is a real vector with v(1:i) = 0 and v(i+1) = 1.

On exit, tau is stored in tau(i), and v(i+2:n) is stored in AP, overwriting A(i+2:n, i).

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 approximate number of floating-point operations is (4/3)n³.

After calling this routine, you can call the following:

opgtr: to form the computed matrix Q explicitly
opmtr: to multiply a real matrix by Q.

The complex counterpart of this routine is hptrd.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?sptrd