nag_dorgbr (f08kfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dorgbr (f08kfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dorgbr (f08kfc) generates one of the real orthogonal matrices Q or PT which were determined by nag_dgebrd (f08kec) when reducing a real matrix to bidiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dorgbr (Nag_OrderType order, Nag_VectType vect, Integer m, Integer n, Integer k, double a[], Integer pda, const double tau[], NagError *fail)

3  Description

nag_dorgbr (f08kfc) is intended to be used after a call to nag_dgebrd (f08kec), which reduces a real rectangular matrix A to bidiagonal form B by an orthogonal transformation: A=QBPT. nag_dgebrd (f08kec) represents the matrices Q and PT as products of elementary reflectors.
This function may be used to generate Q or PT explicitly as square matrices, or in some cases just the leading columns of Q or the leading rows of PT.
The various possibilities are specified by the arguments vect, m, n and k. The appropriate values to cover the most likely cases are as follows (assuming that A was an m by n matrix):
  1. To form the full m by m matrix Q:
    nag_dorgbr(order,Nag_FormQ,m,m,n,...)
    
    (note that the array a must have at least m columns).
  2. If m>n, to form the n leading columns of Q:
    nag_dorgbr(order,Nag_FormQ,m,n,n,...)
    
  3. To form the full n by n matrix PT:
    nag_dorgbr(order,Nag_FormP,n,n,m,...)
    
    (note that the array a must have at least n rows).
  4. If m<n, to form the m leading rows of PT:
    nag_dorgbr(order,Nag_FormP,m,n,m,...)
    

4  References

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

5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     vectNag_VectTypeInput
On entry: indicates whether the orthogonal matrix Q or PT is generated.
vect=Nag_FormQ
Q is generated.
vect=Nag_FormP
PT is generated.
Constraint: vect=Nag_FormQ or Nag_FormP.
3:     mIntegerInput
On entry: m, the number of rows of the orthogonal matrix Q or PT to be returned.
Constraint: m0.
4:     nIntegerInput
On entry: n, the number of columns of the matrix Q or PT to be returned.
Constraints:
  • n0;
  • if vect=Nag_FormQ and m>k, mnk;
  • if vect=Nag_FormQ and mk, m=n;
  • if vect=Nag_FormP and n>k, nmk;
  • if vect=Nag_FormP and nk, n=m.
5:     kIntegerInput
On entry: if vect=Nag_FormQ, the number of columns in the original matrix A.
If vect=Nag_FormP, the number of rows in the original matrix A.
Constraint: k0.
6:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgebrd (f08kec).
On exit: the orthogonal matrix Q or PT, or the leading rows or columns thereof, as specified by vect, m and n.
If order=Nag_ColMajor, the i,jth element of the matrix is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, the i,jth element of the matrix is stored in a[i-1×pda+j-1].
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax1,m;
  • if order=Nag_RowMajor, pdamax1,n.
8:     tau[dim]const doubleInput
Note: the dimension, dim, of the array tau must be at least
  • max1,minm,k when vect=Nag_FormQ;
  • max1,minn,k when vect=Nag_FormP.
On entry: further details of the elementary reflectors, as returned by nag_dgebrd (f08kec) in its argument tauq if vect=Nag_FormQ, or in its argument taup if vect=Nag_FormP.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_3
On entry, vect=value, m=value, n=value and k=value.
Constraint: n0 and
if vect=Nag_FormQ and m>k, mnk;
if vect=Nag_FormQ and mk, m=n;
if vect=Nag_FormP and n>k, nmk;
if vect=Nag_FormP and nk, n=m.
NE_INT
On entry, k=value.
Constraint: k0.
On entry, m=value.
Constraint: m0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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.

7  Accuracy

The computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that
E2 = Oε ,
where ε is the machine precision. A similar statement holds for the computed matrix PT.

8  Further Comments

The total number of floating point operations for the cases listed in Section 3 are approximately as follows:
  1. To form the whole of Q:
    • 43n3m2-3mn+n2 if m>n,
    • 43m3 if mn;
  2. To form the n leading columns of Q when m>n:
    • 23 n2 3m-n ;
  3. To form the whole of PT:
    • 43n3 if mn,
    • 43m3n2-3mn+m2 if m<n;
  4. To form the m leading rows of PT when m<n:
    • 23 m2 3n-m .
The complex analogue of this function is nag_zungbr (f08ktc).

9  Example

For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix A, where
A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50
in the first example and
A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50
in the second. A must first be reduced to tridiagonal form by nag_dgebrd (f08kec). The program then calls nag_dorgbr (f08kfc) twice to form Q and PT, and passes these matrices to nag_dbdsqr (f08mec), which computes the singular value decomposition of A.

9.1  Program Text

Program Text (f08kfce.c)

9.2  Program Data

Program Data (f08kfce.d)

9.3  Program Results

Program Results (f08kfce.r)


nag_dorgbr (f08kfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012