nag_zgebrd (f08ksc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgebrd (f08ksc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgebrd (f08ksc) reduces a complex m by n matrix to bidiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgebrd (Nag_OrderType order, Integer m, Integer n, Complex a[], Integer pda, double d[], double e[], Complex tauq[], Complex taup[], NagError *fail)

3  Description

nag_zgebrd (f08ksc) reduces a complex m by n matrix A to real bidiagonal form B by a unitary transformation: A=QBPH, where Q and PH are unitary matrices of order m and n respectively.
If mn, the reduction is given by:
A =Q B1 0 PH = Q1 B1 PH ,
where B1 is a real n by n upper bidiagonal matrix and Q1 consists of the first n columns of Q.
If m<n, the reduction is given by
A =Q B1 0 PH = Q B1 P1H ,
where B1 is a real m by m lower bidiagonal matrix and P1H consists of the first m rows of PH.
The unitary matrices Q and P are not formed explicitly but are represented as products of elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with Q and P in this representation (see Section 8).

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:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     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.
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 m by n matrix A.
On exit: if mn, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix B, elements below the diagonal are overwritten by details of the unitary matrix Q and elements above the first superdiagonal are overwritten by details of the unitary matrix P.
If m<n, the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix B, elements below the first subdiagonal are overwritten by details of the unitary matrix Q and elements above the diagonal are overwritten by details of the unitary matrix P.
5:     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.
6:     d[dim]doubleOutput
Note: the dimension, dim, of the array d must be at least max1,minm,n.
On exit: the diagonal elements of the bidiagonal matrix B.
7:     e[dim]doubleOutput
Note: the dimension, dim, of the array e must be at least max1,minm,n-1.
On exit: the off-diagonal elements of the bidiagonal matrix B.
8:     tauq[dim]ComplexOutput
Note: the dimension, dim, of the array tauq must be at least max1,minm,n.
On exit: further details of the matrix Q.
9:     taup[dim]ComplexOutput
Note: the dimension, dim, of the array taup must be at least max1,minm,n.
On exit: further details of the matrix P.
10:   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_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
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 bidiagonal form B satisfies QBPH=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 B 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 singular values and vectors.

8  Further Comments

The total number of real floating point operations is approximately 16n23m-n/3 if mn or 16m23n-m/3 if m<n.
If mn, it can be more efficient to first call nag_zgeqrf (f08asc) to perform a QR factorization of A, and then to call nag_zgebrd (f08ksc) to reduce the factor R to bidiagonal form. This requires approximately 8n2m+n floating point operations.
If mn, it can be more efficient to first call nag_zgelqf (f08avc) to perform an LQ factorization of A, and then to call nag_zgebrd (f08ksc) to reduce the factor L to bidiagonal form. This requires approximately 8m2m+n operations.
To form the unitary matrices PH and/or Q nag_zgebrd (f08ksc) may be followed by calls to nag_zungbr (f08ktc):
to form the m by m unitary matrix Q 
nag_zungbr(order,Nag_FormQ,m,m,n,&a,pda,tauq,&fail)
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_zgebrd (f08ksc);
to form the n by n unitary matrix PH 
nag_zungbr(order,Nag_FormP,n,n,m,&a,pda,taup,&fail)
but note that the first dimension of the array a, specified by the argument pda, must be at least n, which may be larger than was required by nag_zgebrd (f08ksc).
To apply Q or P to a complex rectangular matrix C, nag_zgebrd (f08ksc) may be followed by a call to nag_zunmbr (f08kuc).
The real analogue of this function is nag_dgebrd (f08kec).

9  Example

This example reduces the matrix A to bidiagonal form, 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 .

9.1  Program Text

Program Text (f08ksce.c)

9.2  Program Data

Program Data (f08ksce.d)

9.3  Program Results

Program Results (f08ksce.r)


nag_zgebrd (f08ksc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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