?gehrd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?gehrd

Reduces a general matrix to upper Hessenberg form.

Syntax

lapack_int LAPACKE_sgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int ihi, float* a, lapack_int lda, float* tau);

lapack_int LAPACKE_dgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int ihi, double* a, lapack_int lda, double* tau);

lapack_int LAPACKE_cgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int ihi, lapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);

lapack_int LAPACKE_zgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int ihi, lapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);

Include Files

mkl.h

Description

The routine reduces a general matrix A to upper Hessenberg form H by an orthogonal or unitary similarity transformation A = Q*H*Q^H. Here H has real subdiagonal elements.

The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of elementary reflectors. Routines are provided to work with Q in this representation.

Input Parameters

matrix_layout

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

n

The order of the matrix A (n≥ 0).

ilo, ihi

If A is an output by ?gebal, then ilo and ihi must contain the values returned by that routine. Otherwise ilo = 1 and ihi = n. (If n > 0, then 1 ≤ilo≤ihi≤n; if n = 0, ilo = 1 and ihi = 0.)

a

Arrays:

a (size max(1, lda*n)) contains the matrix A.

lda

The leading dimension of a; at least max(1, n).

Output Parameters

a

The elements on and above the subdiagonal contain the upper Hessenberg matrix H. The subdiagonal elements of H are real. The elements below the subdiagonal, with the array tau, represent the orthogonal matrix Q as a product of n elementary reflectors.

tau

Array, size at least max (1, n-1).

Contains scalars that define elementary reflectors for the matrix 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 Hessenberg matrix H is exactly similar to a nearby 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 for real flavors is (2/3)*(ihi - ilo)²(2ihi + 2ilo + 3n); for complex flavors it is 4 times greater.

Parent topic: Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?gehrd