nag_zungbr (f08ktc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zungbr (f08ktc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zungbr (f08ktc) generates one of the complex unitary matrices Q or PH which were determined by nag_zgebrd (f08ksc) when reducing a complex matrix to bidiagonal form.

2  Specification

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

3  Description

nag_zungbr (f08ktc) is intended to be used after a call to nag_zgebrd (f08ksc), which reduces a complex rectangular matrix A to real bidiagonal form B by a unitary transformation: A=QBPH. nag_zgebrd (f08ksc) represents the matrices Q and PH as products of elementary reflectors.
This function may be used to generate Q or PH explicitly as square matrices, or in some cases just the leading columns of Q or the leading rows of PH.
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_zungbr(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_zungbr(order,Nag_FormQ,m,n,n,...)
    
  3. To form the full n by n matrix PH:
    nag_zungbr(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 PH:
    nag_zungbr(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 unitary matrix Q or PH is generated.
vect=Nag_FormQ
Q is generated.
vect=Nag_FormP
PH is generated.
Constraint: vect=Nag_FormQ or Nag_FormP.
3:     mIntegerInput
On entry: m, the number of rows of the unitary matrix Q or PH to be returned.
Constraint: m0.
4:     nIntegerInput
On entry: n, the number of columns of the matrix Q or PH 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]ComplexInput/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_zgebrd (f08ksc).
On exit: the unitary matrix Q or PH, 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) of the matrix A in the array a.
Constraint: pdamax1,m.
8:     tau[dim]const ComplexInput
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_zgebrd (f08ksc) 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.
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 unitary matrix by a matrix E such that
E2 = Oε ,
where ε is the machine precision. A similar statement holds for the computed matrix PH.

8  Further Comments

The total number of real floating point operations for the cases listed in Section 3 are approximately as follows:
  1. To form the whole of Q:
    • 163n3m2-3mn+n2 if m>n,
    • 163m3 if mn;
  2. To form the n leading columns of Q when m>n:
    • 83n23m-n;
  3. To form the whole of PH:
    • 163n3 if mn,
    • 163m33n2-3mn+m2 if m<n;
  4. To form the m leading rows of PH when m<n:
    • 83m23n-m.
The real analogue of this function is nag_dorgbr (f08kfc).

9  Example

For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix A, where
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i
in the first example and
A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i
in the second. A must first be reduced to tridiagonal form by nag_zgebrd (f08ksc). The program then calls nag_zungbr (f08ktc) twice to form Q and PH, and passes these matrices to nag_zbdsqr (f08msc), which computes the singular value decomposition of A.

9.1  Program Text

Program Text (f08ktce.c)

9.2  Program Data

Program Data (f08ktce.d)

9.3  Program Results

Program Results (f08ktce.r)


nag_zungbr (f08ktc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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