NAG Library Function Document

nag_zgghd3 (f08wtc)

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1
Purpose

nag_zgghd3 (f08wtc) reduces a pair of complex matrices A,B, where B is upper triangular, to the generalized upper Hessenberg form using unitary transformations.

2
Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgghd3 (Nag_OrderType order, Nag_ComputeQType compq, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, Complex a[], Integer pda, Complex b[], Integer pdb, Complex q[], Integer pdq, Complex z[], Integer pdz, NagError *fail)

3
Description

nag_zgghd3 (f08wtc) is usually the third step in the solution of the complex generalized eigenvalue problem
Ax=λBx.  
The (optional) first step balances the two matrices using nag_zggbal (f08wvc). In the second step, matrix B is reduced to upper triangular form using the QR factorization function nag_zgeqrf (f08asc) and this unitary transformation Q is applied to matrix A by calling nag_zunmqr (f08auc). The driver, nag_zggev3 (f08wqc), solves the complex generalized eigenvalue problem by combining all the required steps including those just listed.
nag_zgghd3 (f08wtc) reduces a pair of complex matrices A,B, where B is triangular, to the generalized upper Hessenberg form using unitary transformations. This two-sided transformation is of the form
QHAZ=H, QHBZ=T  
where H is an upper Hessenberg matrix, T is an upper triangular matrix and Q and Z are unitary matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that
Q1AZ1H=Q1QHZ1ZH, Q1BZ1H=Q1QTZ1ZH.  

4
References

Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

5
Arguments

1:     order Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     compq Nag_ComputeQTypeInput
On entry: specifies the form of the computed unitary matrix Q.
compq=Nag_NotQ
Do not compute Q.
compq=Nag_InitQ
The unitary matrix Q is returned.
compq=Nag_UpdateSchur
q must contain a unitary matrix Q1, and the product Q1Q is returned.
Constraint: compq=Nag_NotQ, Nag_InitQ or Nag_UpdateSchur.
3:     compz Nag_ComputeZTypeInput
On entry: specifies the form of the computed unitary matrix Z.
compz=Nag_NotZ
Do not compute Z.
compz=Nag_UpdateZ
z must contain a unitary matrix Z1, and the product Z1Z is returned.
compz=Nag_InitZ
The unitary matrix Z is returned.
Constraint: compz=Nag_NotZ, Nag_UpdateZ or Nag_InitZ.
4:     n IntegerInput
On entry: n, the order of the matrices A and B.
Constraint: n0.
5:     ilo IntegerInput
6:     ihi IntegerInput
On entry: ilo and ihi as determined by a previous call to nag_zggbal (f08wvc). Otherwise, they should be set to 1 and n, respectively.
Constraints:
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
7:     a[dim] ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth 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 matrix A of the matrix pair A,B. Usually, this is the matrix A returned by nag_zunmqr (f08auc).
On exit: a is overwritten by the upper Hessenberg matrix H.
8:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
9:     b[dim] ComplexInput/Output
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the upper triangular matrix B of the matrix pair A,B. Usually, this is the matrix B returned by the QR factorization function nag_zgeqrf (f08asc).
On exit: b is overwritten by the upper triangular matrix T.
10:   pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
11:   q[dim] ComplexInput/Output
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when compq=Nag_InitQ or Nag_UpdateSchur;
  • 1 when compq=Nag_NotQ.
The i,jth element of the matrix Q is stored in
  • q[j-1×pdq+i-1] when order=Nag_ColMajor;
  • q[i-1×pdq+j-1] when order=Nag_RowMajor.
On entry: if compq=Nag_UpdateSchur, q must contain a unitary matrix Q1.
If compq=Nag_NotQ, q is not referenced.
On exit: if compq=Nag_InitQ, q contains the unitary matrix Q.
Iif compq=Nag_UpdateSchur, q is overwritten by Q1Q.
12:   pdq IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if compq=Nag_InitQ or Nag_UpdateSchur, pdq max1,n ;
  • if compq=Nag_NotQ, pdq1.
13:   z[dim] ComplexInput/Output
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when compz=Nag_UpdateZ or Nag_InitZ;
  • 1 when compz=Nag_NotZ.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On entry: if compz=Nag_UpdateZ, z must contain a unitary matrix Z1.
If compz=Nag_NotZ, z is not referenced.
On exit: if compz=Nag_InitZ, z contains the unitary matrix Z.
If compz=Nag_UpdateZ, z is overwritten by Z1Z.
14:   pdz IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if compz=Nag_UpdateZ or Nag_InitZ, pdz max1,n ;
  • if compz=Nag_NotZ, pdz1.
15:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_2
On entry, compq=value, pdq=value and n=value.
Constraint: if compq=Nag_InitQ or Nag_UpdateSchur, pdq max1,n ;
if compq=Nag_NotQ, pdq1.
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_UpdateZ or Nag_InitZ, pdz max1,n ;
if compz=Nag_NotZ, pdz1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
NE_INT_3
On entry, n=value, ilo=value and ihi=value.
Constraint: ilo1 and
if n>0, 1 ilo ihi n ;
if n=0, ilo=1 and ihi=0.
NE_INTERNAL_ERROR
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

The reduction to the generalized Hessenberg form is implemented using unitary transformations which are backward stable.

8
Parallelism and Performance

nag_zgghd3 (f08wtc) 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

This function is usually followed by nag_zhgeqz (f08xsc) which implements the QZ algorithm for computing generalized eigenvalues of a reduced pair of matrices.
The real analogue of this function is nag_dgghd3 (f08wfc).

10
Example

See Section 10 in nag_zhgeqz (f08xsc) and nag_ztgevc (f08yxc).
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017