Visible to Intel only — GUID: GUID-0CA713FB-F989-4B3F-9EB9-8244480848D7
Visible to Intel only — GUID: GUID-0CA713FB-F989-4B3F-9EB9-8244480848D7
?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*QH. 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.