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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgeqrt (f08ab)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dgeqrt (f08ab) recursively computes, with explicit blocking, the QR factorization of a real m by n matrix.


[a, t, info] = f08ab(nb, a, 'm', m, 'n', n)
[a, t, info] = nag_lapack_dgeqrt(nb, a, 'm', m, 'n', n)


nag_lapack_dgeqrt (f08ab) forms the QR factorization of an arbitrary rectangular real m by n matrix. No pivoting is performed.
It differs from nag_lapack_dgeqrf (f08ae) in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the QR factorization based on the algorithm of Elmroth and Gustavson (2000).
If mn, the factorization is given by:
A = Q R 0 ,  
where R is an n by n upper triangular matrix and Q is an m by m orthogonal matrix. It is sometimes more convenient to write the factorization as
A = Q1 Q2 R 0 ,  
which reduces to
A = Q1 R ,  
where Q1 consists of the first n columns of Q, and Q2 the remaining m-n columns.
If m<n, R is upper 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 represents a QR factorization of the first k columns of the original matrix A.


Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel QR Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44) 4 605–624
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore


Compulsory Input Parameters

1:     nb int64int32nag_int scalar
The explicitly chosen block size to be used in computing the QR factorization. See Further Comments for details.
  • nb1;
  • if minm,n>0, nbminm,n.
2:     alda: – double 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: – double 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 orthogonal 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 orthogonal matrix Q and the remaining elements store the corresponding elements of the m by n upper trapezoidal matrix R.
2:     tldt: – double array
The first dimension of the array t will be nb.
The second dimension of the array t will be max1,minm,n.
Further details of the orthogonal matrix Q. 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 each of the blocks, an upper triangular block reflector factor is computed: T1,T2,,Tb. These are stored in the nb by n matrix T as T=T1|T2||Tb.
3:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.


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 floating-point operations is approximately 23 n2 3m-n  if mn or 23 m2 3n-m  if m<n.
To apply Q to an arbitrary real rectangular matrix C, nag_lapack_dgeqrt (f08ab) may be followed by a call to nag_lapack_dgemqrt (f08ac). For example,
[t, c, info] = f08ac('Left', 'Transpose', nb, a, t, c, 'k', min(m,n));
forms C=QTC, where C is m by p.
To form the orthogonal matrix Q explicitly, simply initialize the m by m matrix C to the identity matrix and form C=QC using nag_lapack_dgemqrt (f08ac) as above.
The block size, nb, used by nag_lapack_dgeqrt (f08ab) is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of nb=64minm,n is likely to achieve good efficiency and it is unlikely that an optimal value would exceed 340.
To compute a QR factorization with column pivoting, use nag_lapack_dtpqrt (f08bb) or nag_lapack_dgeqpf (f08be).
The complex analogue of this function is nag_lapack_zgeqrt (f08ap).


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 = -0.57 -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   and   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 .  
function f08ab_example

fprintf('f08ab 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');

% 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));
fprintf('Square roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);

f08ab 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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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