NAG CL Interface
f08nsc (zgehrd)

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1 Purpose

f08nsc reduces a complex general matrix to Hessenberg form.

2 Specification

#include <nag.h>
void  f08nsc (Nag_OrderType order, Integer n, Integer ilo, Integer ihi, Complex a[], Integer pda, Complex tau[], NagError *fail)
The function may be called by the names: f08nsc, nag_lapackeig_zgehrd or nag_zgehrd.

3 Description

f08nsc reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: A=QHQH. H has real subdiagonal elements.
The matrix Q is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Section 9).
The function can take advantage of a previous call to f08nvc, which may produce a matrix with the structure:
( A11 A12 A13 A22 A23 A33 )  
where A11 and A33 are upper triangular. If so, only the central diagonal block A22, in rows and columns ilo to ihi, needs to be reduced to Hessenberg form (the blocks A12 and A23 will also be affected by the reduction). Therefore, the values of ilo and ihi determined by f08nvc can be supplied to the function directly. If f08nvc has not previously been called however, then ilo must be set to 1 and ihi to n.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: ilo Integer Input
4: ihi Integer Input
On entry: if A has been output by f08nvc, ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to 1 and ihi to n.
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
5: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the n×n general matrix A.
On exit: a is overwritten by the upper Hessenberg matrix H and details of the unitary matrix Q. The subdiagonal elements of H are real.
6: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
7: tau[dim] Complex Output
Note: the dimension, dim, of the array tau must be at least max(1,n-1).
On exit: further details of the unitary matrix Q.
8: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, n=value, ilo=value and ihi=value.
Constraint: if n>0, 1 ilo ihi n ;
if n=0, ilo=1 and ihi=0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed Hessenberg matrix H is exactly similar to a nearby matrix (A+E), where
E2 c (n) ε A2 ,  
c(n) is a modestly increasing function of n, and ε is the machine precision.
The elements of H themselves may be sensitive to small perturbations in A or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08nsc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately 83q2(2q+3n), where q=ihi-ilo; if ilo=1 and ihi=n, the number is approximately 403n3.
To form the unitary matrix Q f08nsc may be followed by a call to f08ntc :
To apply Q to an m×n complex matrix C f08nsc may be followed by a call to f08nuc. For example,
forms the matrix product QC.
The real analogue of this function is f08nec.

10 Example

This example computes the upper Hessenberg form of the matrix A, where
A = ( -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i ) .  

10.1 Program Text

Program Text (f08nsce.c)

10.2 Program Data

Program Data (f08nsce.d)

10.3 Program Results

Program Results (f08nsce.r)