NAG Library Function Document

nag_dgehrd (f08nec)


    1  Purpose
    7  Accuracy


nag_dgehrd (f08nec) reduces a real general matrix to Hessenberg form.


#include <nag.h>
#include <nagf08.h>
void  nag_dgehrd (Nag_OrderType order, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, double tau[], NagError *fail)


nag_dgehrd (f08nec) reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT.
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 nag_dgebal (f08nhc), 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 nag_dgebal (f08nhc) can be supplied to the function directly. If nag_dgebal (f08nhc) has not previously been called however, then ilo must be set to 1 and ihi to n.


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


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 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:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     ilo IntegerInput
4:     ihi IntegerInput
On entry: if A has been output by nag_dgebal (f08nhc), 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] doubleInput/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 n by n general matrix A.
On exit: a is overwritten by the upper Hessenberg matrix H and details of the orthogonal matrix Q.
6:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     tau[dim] doubleOutput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On exit: further details of the orthogonal matrix Q.
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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: pdamax1,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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.


The computed Hessenberg matrix H is exactly similar to a nearby matrix A+E, where
E2 c n ε A2 ,  
cn 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.

Parallelism and Performance

nag_dgehrd (f08nec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgehrd (f08nec) 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.

Further Comments

The total number of floating-point operations is approximately 23q22q+3n, where q=ihi-ilo; if ilo=1 and ihi=n, the number is approximately 103n3.
To form the orthogonal matrix Q nag_dgehrd (f08nec) may be followed by a call to nag_dorghr (f08nfc):
To apply Q to an m by n real matrix C nag_dgehrd (f08nec) may be followed by a call to nag_dormhr (f08ngc). For example,
forms the matrix product QC.
The complex analogue of this function is nag_zgehrd (f08nsc).


This example computes the upper Hessenberg form of the matrix A, where
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .  

Program Text

Program Text (f08nece.c)

Program Data

Program Data (f08nece.d)

Program Results

Program Results (f08nece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017