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*QT. 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*vT

    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*vT

    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.

Para obtener información más completa sobre las optimizaciones del compilador, consulte nuestro Aviso de optimización.
Seleccione el color del botón adhesivo: 
Orange (only for download buttons)