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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgeqrt (f08ap)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zgeqrt (f08ap) recursively computes, with explicit blocking, the QR factorization of a complex m by n matrix.


[a, t, info] = f08ap(nb, a, 'm', m, 'n', n)
[a, t, info] = nag_lapack_zgeqrt(nb, a, 'm', m, 'n', n)


nag_lapack_zgeqrt (f08ap) forms the QR factorization of an arbitrary rectangular complex m by n matrix. No pivoting is performed.
It differs from nag_lapack_zgeqrf (f08as) 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 (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 = 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: – 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:     tldt: – complex 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 unitary 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 real floating-point operations is approximately 83 n2 3m-n  if mn or 83 m2 3n-m  if m<n.
To apply Q to an arbitrary complex rectangular matrix C, nag_lapack_zgeqrt (f08ap) may be followed by a call to nag_lapack_zgemqrt (f08aq). For example,
[t, c, info] = f08aq('Left', 'Conjugate Transpose', nb, a, t, c);
forms C=QHC, where C is m by p.
To form the unitary matrix Q explicitly, simply initialize the m by m matrix C to the identity matrix and form C=QC using nag_lapack_zgemqrt (f08aq) as above.
The block size, nb, used by nag_lapack_zgeqrt (f08ap) 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_ztpqrt (f08bp) or nag_lapack_zgeqpf (f08bs).
The real analogue of this function is nag_lapack_dgeqrt (f08ab).


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  
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 .  
function f08ap_example

fprintf('f08ap 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 = [ 0.96 - 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 Factorisation of A
[QR, T, info] = f08ap(n,a);

% Compute C = (C1) = (Q^H)*B
[c1, info] = f08aq(...
                  'Left', 'Conjugate Transpose', QR, T, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07ts(...
                  '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('\nSquare roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);

f08ap example results

Least-squares solutions
  -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 roots of the residual sums of squares
    6.88e-02    1.87e-01

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

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