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NAG Toolbox: nag_lapack_zgeqrf (f08as)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgeqrf (f08as) computes the QR factorization of a complex m by n matrix.

Syntax

[a, tau, info] = f08as(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_zgeqrf(a, 'm', m, 'n', n)

Description

nag_lapack_zgeqrf (f08as) forms the QR factorization of an arbitrary rectangular complex m by n matrix. No pivoting is performed.
If mn, the factorization is given by:
A = Q R 0 ,  
where R is an n by n upper triangular matrix (with real diagonal elements) and Q is an m by m unitary matrix. It is sometimes more convenient to write the factorization as
A = Q1 Q2 R 0 ,  
which reduces to
A = Q1R ,  
where Q1 consists of the first n columns of Q, and Q2 the remaining m-n columns.
If m<n, R is trapezoidal, and the factorization can be written
A = Q R1 R2 ,  
where R1 is upper triangular and R2 is rectangular.
The matrix Q is not formed explicitly but is represented as a product of minm,n elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Further Comments).
Note also that for any k<n, the information returned in the first k columns of the array a represents a QR factorization of the first k columns of the original matrix A.

References

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

Parameters

Compulsory Input Parameters

1:     alda: – complex array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
The m by n matrix A.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the matrix A.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
If mn, the elements below the diagonal store details of the unitary matrix Q and the upper triangle stores the corresponding elements of the n by n upper triangular matrix R.
If m<n, the strictly lower triangular part stores details of the unitary matrix Q and the remaining elements store the corresponding elements of the m by n upper trapezoidal matrix R.
The diagonal elements of R are real.
2:     tau: – complex array
The dimension of the array tau will be max1,minm,n
Further details of the unitary matrix Q.
3:     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: m, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: 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 factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision.

Further Comments

The total number of real floating-point operations is approximately 83 n2 3m-n  if mn or 83 m2 3n-m  if m<n.
To form the unitary matrix Q nag_lapack_zgeqrf (f08as) may be followed by a call to nag_lapack_zungqr (f08at):
[a, info] = f08at(a, tau);
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_lapack_zgeqrf (f08as).
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
[a, info] = f08at(a(:,1:n), tau);
To apply Q to an arbitrary complex rectangular matrix C, nag_lapack_zgeqrf (f08as) may be followed by a call to nag_lapack_zunmqr (f08au). For example,
[c, info] = f08au('Left', 'Conjugate Transpose', a, tau, c);
forms C=QHC, where C is m by p.
To compute a QR factorization with column pivoting, use nag_lapack_zgeqpf (f08bs).
The real analogue of this function is nag_lapack_dgeqrf (f08ae).

Example

This example solves the linear least squares problems
minimize Axi - bi 2 ,   i=1,2  
where b1 and b2 are the columns of the matrix B,
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  
and
B = -1.54+0.76i 3.17-2.09i 0.12-1.92i -6.53+4.18i -9.08-4.31i 7.28+0.73i 7.49+3.65i 0.91-3.97i -5.63-2.12i -5.46-1.64i 2.37+8.03i -2.84-5.86i .  
function f08as_example


fprintf('f08as example results\n\n');

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];
b = [-2.09 + 1.93i,  3.26 - 2.70i;
      3.34 - 3.53i, -6.22 + 1.16i;
     -4.94 - 2.04i,  7.94 - 3.13i;
      0.17 + 4.23i,  1.04 - 4.26i;
     -5.19 + 3.63i, -2.31 - 2.12i;
      0.98 + 2.53i, -1.39 - 4.05i];
% Compute the QR factorization of A
[qr, tau, info] = f08as(a);

% Compute C = [c1;C2] = (Q^H)*B
[c, info] = f08au(...
                  'Left', 'Conjugate transpose', qr, tau, b);

% Compute least-squares solutions by backsubstitution in R*x = C1
[x, info] = f07ts(...
                  'Upper', 'No transpose', 'Non-Unit', qr(1:4,:), c(1:4,:));

fprintf('Least-squares solution(s)\n');
disp(x);

fprintf('Square root(s) of the residual sum(s) of squares\n');
for i=1:2
  fprintf('%8.3f ',norm(c(5:6,i)));
end
fprintf('\n');


f08as example results

Least-squares solution(s)
  -0.5044 - 1.2179i   0.7629 + 1.4529i
  -2.4281 + 2.8574i   5.1570 - 3.6089i
   1.4872 - 2.1955i  -2.6518 + 2.1203i
   0.4537 + 2.6904i  -2.7606 + 0.3318i

Square root(s) of the residual sum(s) of squares
   0.069    0.187 

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