?sptrd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?sptrd

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

Syntax

lapack_int LAPACKE_ssptrd (int matrix_layout, char uplo, lapack_int n, float* ap, float* d, float* e, float* tau);

lapack_int LAPACKE_dsptrd (int matrix_layout, char uplo, lapack_int n, double* ap, double* d, double* e, double* tau);

Include Files

mkl.h

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

matrix_layout

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

uplo

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

The order of the matrix A (n≥ 0).

ap

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

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).

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 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 - 1], 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 - 1], 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 C

?sptrd