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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zunmbr (f08ku)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_lapack_zunmbr (f08ku) multiplies an arbitrary complex m by n matrix C by one of the complex unitary matrices Q or P which were determined by nag_lapack_zgebrd (f08ks) when reducing a complex matrix to bidiagonal form.

Syntax

[c, info] = f08ku(vect, side, trans, k, a, tau, c, 'm', m, 'n', n)
[c, info] = nag_lapack_zunmbr(vect, side, trans, k, a, tau, c, 'm', m, 'n', n)

Description

nag_lapack_zunmbr (f08ku) is intended to be used after a call to nag_lapack_zgebrd (f08ks), which reduces a complex rectangular matrix A to real bidiagonal form B by a unitary transformation: A=QBPH. nag_lapack_zgebrd (f08ks) represents the matrices Q and PH as products of elementary reflectors.
This function may be used to form one of the matrix products
QC , QHC , CQ , CQH , PC , PHC , CP ​ or ​ CPH ,  
overwriting the result on C (which may be any complex rectangular matrix).

References

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

Parameters

Note: in the descriptions below, r denotes the order of Q or PH: if side='L', r=m and if side='R', r=n.

Compulsory Input Parameters

1:     vect – string (length ≥ 1)
Indicates whether Q or QH or P or PH is to be applied to C.
vect='Q'
Q or QH is applied to C.
vect='P'
P or PH is applied to C.
Constraint: vect='Q' or 'P'.
2:     side – string (length ≥ 1)
Indicates how Q or QH or P or PH is to be applied to C.
side='L'
Q or QH or P or PH is applied to C from the left.
side='R'
Q or QH or P or PH is applied to C from the right.
Constraint: side='L' or 'R'.
3:     trans – string (length ≥ 1)
Indicates whether Q or P or QH or PH is to be applied to C.
trans='N'
Q or P is applied to C.
trans='C'
QH or PH is applied to C.
Constraint: trans='N' or 'C'.
4:     k – int64int32nag_int scalar
If vect='Q', the number of columns in the original matrix A.
If vect='P', the number of rows in the original matrix A.
Constraint: k≥0.
5:     alda: – complex array
The first dimension, lda, of the array a must satisfy
  • if vect='Q', lda≥ max1,r ;
  • if vect='P', lda≥ max1,minr,k .
The second dimension of the array a must be at least max1,minr,k  if vect='Q' and at least max1,r if vect='P'.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgebrd (f08ks).
6:     tau: – complex array
The dimension of the array tau must be at least max1,minr,k
Further details of the elementary reflectors, as returned by nag_lapack_zgebrd (f08ks) in its argument tauq if vect='Q', or in its argument taup if vect='P'.
7:     cldc: – complex array
The first dimension of the array c must be at least max1,m.
The second dimension of the array c must be at least max1,n.
The matrix C.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: the first dimension of the array c.
m, the number of rows of the matrix C.
Constraint: m≥0.
2:     n – int64int32nag_int scalar
Default: the second dimension of the array c.
n, the number of columns of the matrix C.
Constraint: n≥0.

Output Parameters

1:     cldc: – complex array
The first dimension of the array c will be max1,m.
The second dimension of the array c will be max1,n.
c stores QC or QHC or CQ or CHQ or PC or PHC or CP or CHP as specified by vect, side and trans.
2:     info – int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: vect, 2: side, 3: trans, 4: m, 5: n, 6: k, 7: a, 8: lda, 9: tau, 10: c, 11: ldc, 12: work, 13: lwork, 14: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed result differs from the exact result by a matrix E such that
E2 = Oε C2 ,  
where ε is the machine precision.

Further Comments

The total number of real floating-point operations is approximately where k is the value of the argument k
The real analogue of this function is nag_lapack_dormbr (f08kg).

Example

For this function two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix A may be preceded by a QR or LQ factorization of A.
In the first example, m>n, and
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 .  
The function first performs a QR factorization of A as A=QaR and then reduces the factor R to bidiagonal form B: R=QbBPH. Finally it forms Qa and calls nag_lapack_zunmbr (f08ku) to form Q=QaQb.
In the second example, m<n, 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 .  
The function first performs an LQ factorization of A as A=LPaH and then reduces the factor L to bidiagonal form B: L=QBPbH. Finally it forms PbH and calls nag_lapack_zunmbr (f08ku) to form PH=PbHPaH.

Open in the MATLAB editor: f08ku_example

function f08ku_example


fprintf('f08ku example results\n\n');

% Two cases of preceding reduction to bidiagonal form by QR or LQ
% Case 1: m > n, precede by QR
ex1;
% Case 2: m < n, precede by LQ
ex2;



function ex1
  m = int64(6);
  n = int64(4);
  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];

  % Factorize A = QR
  [QR, tau, info] = f08as(a);

  % Generate Q from QR
  [Q, info] = f08at(QR, tau);

  % Extract R from QR
  R = triu(QR(1:n,1:n));

  % Bidiagonalize R = Q1 B P^H
  [B, d, e, tauq, taup, info] = ...
  f08ks(R);

  % Update Q: Q2 = Q*Q1 (so A = QR = Q2 B P^H)
  vect = 'Q';
  side = 'Right';
  trans = 'No transpose';
  [Q2, info] = f08ku( ...
                      vect, side, trans, n, B, tauq, Q);

  fprintf('Example 1: bidiagonal matrix B\n   Main diagonal  ');
  fprintf(' %7.3f',d);
  fprintf('\n   super-diagonal ');
  fprintf(' %7.3f',e);
  fprintf('\n\n');
  disp('Example 1: Orthogonal matrix Q');
  disp(Q2);

function ex2
  m = int64(3);
  n = int64(4);
  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];

  % Factorize A = LQ
  [LQ, tau, info] = f08av(a);

  % Generate Q from LQ
  [Q, info] = f08aw(LQ, tau);

  % Extract L from LQ
  L = tril(LQ(1:m,1:m));

  % Bidiagonalize L = Q1 B P^H
  [B, d, e, tauq, taup, info] = ...
  f08ks(L);

  % Update Q: P2 = P^H*Q (so A = LQ = Q1 B P2)
  vect = 'P';
  side = 'Left';
  trans = 'Conjugate Transpose';
  [P2, info] = f08ku( ...
                      vect, side, trans, n, B, taup, Q);

  fprintf('Example 2: bidiagonal matrix B\n   Main diagonal  ');
  fprintf(' %7.3f',d);
  fprintf('\n   super-diagonal ');
  fprintf(' %7.3f',e);
  fprintf('\n\n');
  disp('Example 2: Orthogonal matrix P^H');
  disp(P2);

f08ku example results

Example 1: bidiagonal matrix B
   Main diagonal    -3.087  -2.066  -1.873  -2.002
   super-diagonal    2.113  -1.263   1.613

Example 1: Orthogonal matrix Q
  -0.3110 + 0.2624i   0.6521 + 0.5532i   0.0427 + 0.0361i  -0.2634 - 0.0741i
   0.3175 - 0.6414i   0.3488 + 0.0721i   0.2287 + 0.0069i   0.1101 - 0.0326i
  -0.2008 + 0.1490i  -0.3103 + 0.0230i   0.1855 - 0.1817i  -0.2956 + 0.5648i
   0.1199 - 0.1231i  -0.0046 - 0.0005i  -0.3305 + 0.4821i  -0.0675 + 0.3464i
  -0.2689 - 0.1652i   0.1794 - 0.0586i  -0.5235 - 0.2580i   0.3927 + 0.1450i
  -0.3499 + 0.0907i   0.0829 - 0.0506i   0.3202 + 0.3038i   0.3174 + 0.3241i

Example 2: bidiagonal matrix B
   Main diagonal     2.761   1.630  -1.327
   super-diagonal   -0.950  -1.018

Example 2: Orthogonal matrix P^H
  -0.1258 + 0.1618i  -0.2247 + 0.3864i   0.3460 + 0.2157i  -0.7099 - 0.2966i
   0.4148 + 0.1795i   0.1368 - 0.3976i   0.6885 + 0.3386i   0.1667 - 0.0494i
   0.4575 - 0.4807i  -0.2733 + 0.4981i  -0.0230 + 0.3861i   0.1730 + 0.2395i
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