?gehrd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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?gehrd

Reduces a general matrix to upper Hessenberg form.

Syntax

call sgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info)

call dgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info)

call cgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info)

call zgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info)

call gehrd(a [, tau] [,ilo] [,ihi] [,info])

Include Files

mkl.fi, lapack.f90

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

n

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

ilo, ihi

INTEGER. 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, work

REAL for sgehrd

DOUBLE PRECISION for dgehrd

COMPLEX for cgehrd

DOUBLE COMPLEX for zgehrd.

Arrays:

a(lda,*) contains the matrix A.

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

work (lwork) is a workspace array.

lda

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

lwork

INTEGER. The size of the work array; at least max(1, n).

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

See Application Notes for the suggested value of lwork.

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

REAL for sgehrd

DOUBLE PRECISION for dgehrd

COMPLEX for cgehrd

DOUBLE COMPLEX for zgehrd.

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

Contains scalars that define elementary reflectors for the matrix Q.

work(1)

If info = 0, on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

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 gehrd interface are the following:

a: Holds the matrix A of size (n,n).
tau: Holds the vector of length (n-1).
ilo: Default value for this argument is ilo = 1.
ihi: Default value for this argument is ihi = n.

Application Notes

For better performance, try using lwork = n*blocksize, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

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 Fortran

?gehrd

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes