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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgemqrt (f08ac)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgemqrt (f08ac) multiplies an arbitrary real matrix C by the real orthogonal matrix Q from a QR factorization computed by nag_lapack_dgeqrt (f08ab).

Syntax

[c, info] = f08ac(side, trans, v, t, c, 'm', m, 'n', n, 'k', k, 'nb', nb)
[c, info] = nag_lapack_dgemqrt(side, trans, v, t, c, 'm', m, 'n', n, 'k', k, 'nb', nb)

Description

nag_lapack_dgemqrt (f08ac) is intended to be used after a call to nag_lapack_dgeqrt (f08ab) which performs a QR factorization of a real matrix A. The orthogonal matrix Q is represented as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC , QTC , CQ ​ or ​ CQT ,  
overwriting the result on C (which may be any real rectangular matrix).
A common application of this function is in solving linear least squares problems, as described in the F08 Chapter Introduction and illustrated in Example in nag_lapack_dgeqrt (f08ab).

References

Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     side – string (length ≥ 1)
Indicates how Q or QT is to be applied to C.
side='L'
Q or QT is applied to C from the left.
side='R'
Q or QT is applied to C from the right.
Constraint: side='L' or 'R'.
2:     trans – string (length ≥ 1)
Indicates whether Q or QT is to be applied to C.
trans='N'
Q is applied to C.
trans='T'
QT is applied to C.
Constraint: trans='N' or 'T'.
3:     vldv: – double array
The first dimension, ldv, of the array v must satisfy
  • if side='L', ldv max1,m ;
  • if side='R', ldv max1,n .
The second dimension of the array v must be at least max1,k.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgeqrt (f08ab) in the first k columns of its array argument a.
4:     tldt: – double array
The first dimension of the array t must be at least nb.
The second dimension of the array t must be at least max1,k.
Further details of the orthogonal matrix Q as returned by nag_lapack_dgeqrt (f08ab). The number of blocks is b=knb, where k=minm,n and each block is of order nb except for the last block, which is of order k-b-1×nb. For the b blocks the upper triangular block reflector factors T1,T2,,Tb are stored in the nb by n matrix T as T=T1|T2||Tb.
5:     cldc: – double 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 m by n 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: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array c.
n, the number of columns of the matrix C.
Constraint: n0.
3:     k int64int32nag_int scalar
Default: the second dimension of the arrays v, t.
k, the number of elementary reflectors whose product defines the matrix Q. Usually k=minmA,nA where mA, nA are the dimensions of the matrix A supplied in a previous call to nag_lapack_dgeqrt (f08ab).
Constraints:
  • if side='L', m k 0 ;
  • if side='R', n k 0 .
4:     nb int64int32nag_int scalar
Default: the first dimension of the array t.
The block size used in the QR factorization performed in a previous call to nag_lapack_dgeqrt (f08ab); this value must remain unchanged from that call.
Constraints:
  • nb1;
  • if k>0, nbk.

Output Parameters

1:     cldc: – double 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 QTC or CQ or CQT as specified by 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<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

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 floating-point operations is approximately 2nk 2m-k  if side='L' and 2mk 2n-k  if side='R'.
The complex analogue of this function is nag_lapack_zgemqrt (f08aq).

Example

See Example in nag_lapack_dgeqrt (f08ab).
function f08ac_example


fprintf('f08ac example results\n\n');

% Minimize ||Ax - b|| using recursive QR for m-by-n A and m-by-p B

m = int64(6);
n = int64(4);
p = int64(2);
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];
b = [-2.67,  0.41;
     -0.55, -3.10;
      3.34, -4.01;
     -0.77,  2.76;
      0.48, -6.17;
      4.10,  0.21];

% Compute the QR Factorisation of A
[QR, T, info] = f08ab(n,a);

% Compute C = (C1) = (Q^T)*B
[c1, info] = f08ac(...
                  'Left', 'Transpose', QR, T, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07te(...
                  'Upper', 'No Transpose', 'Non-Unit', QR, c1, 'n', n);

% Print least-squares solutions
disp('Least-squares solutions');
disp(x(1:n,:));

% Compute and print estimates of the square roots of the residual
% sums of squares
for j=1:p
  rnorm(j) = norm(x(n+1:m,j));
end
fprintf('\nSquare roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);
fprintf('\n');


f08ac example results

Least-squares solutions
    1.5339   -1.5753
    1.8707    0.5559
   -1.5241    1.3119
    0.0392    2.9585


Square roots of the residual sums of squares
    2.22e-02    1.38e-02

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