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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztzrzf (f08bv)

## Purpose

nag_lapack_ztzrzf (f08bv) reduces the $m$ by $n$ ($m\le n$) complex upper trapezoidal matrix $A$ to upper triangular form by means of unitary transformations.

## Syntax

[a, tau, info] = f08bv(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_ztzrzf(a, 'm', m, 'n', n)

## Description

The $m$ by $n$ ($m\le n$) complex upper trapezoidal matrix $A$ given by
 $A = R1 R2 ,$
where ${R}_{1}$ is an $m$ by $m$ upper triangular matrix and ${R}_{2}$ is an $m$ by $\left(n-m\right)$ matrix, is factorized as
 $A = R 0 Z ,$
where $R$ is also an $m$ by $m$ upper triangular matrix and $Z$ is an $n$ by $n$ unitary matrix.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The leading $m$ by $n$ upper trapezoidal part of the array a must contain the matrix to be factorized.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The leading $m$ by $m$ upper triangular part of a contains the upper triangular matrix $R$, and elements ${\mathbf{m}}+1$ to n of the first $m$ rows of a, with the array tau, represent the unitary matrix $Z$ as a product of $m$ elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The scalar factors of the elementary reflectors.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, 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
 $E2 = O⁡ε A2$
and $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $16{m}^{2}\left(n-m\right)$.
The real analogue of this function is nag_lapack_dtzrzf (f08bh).

## Example

This example solves the linear least squares problems
 $minx bj - Axj 2 , j=1,2$
for the minimum norm solutions ${x}_{1}$ and ${x}_{2}$, where ${b}_{j}$ is the $j$th column of the matrix $B$,
 $A = 0.47-0.34i -0.40+0.54i 0.60+0.01i 0.80-1.02i -0.32-0.23i -0.05+0.20i -0.26-0.44i -0.43+0.17i 0.35-0.60i -0.52-0.34i 0.87-0.11i -0.34-0.09i 0.89+0.71i -0.45-0.45i -0.02-0.57i 1.14-0.78i -0.19+0.06i 0.11-0.85i 1.44+0.80i 0.07+1.14i$
and
 $B = -1.08-2.59i 2.22+2.35i -2.61-1.49i 1.62-1.48i 3.13-3.61i 1.65+3.43i 7.33-8.01i -0.98+3.08i 9.12+7.63i -2.84+2.78i .$
The solution is obtained by first obtaining a $QR$ factorization with column pivoting of the matrix $A$, and then the $RZ$ factorization of the leading $k$ by $k$ part of $R$ is computed, where $k$ is the estimated rank of $A$. A tolerance of $0.01$ is used to estimate the rank of $A$ from the upper triangular factor, $R$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
function f08bv_example

fprintf('f08bv example results\n\n');

% Find Least squares solution of Ax = B, m>n
m = 5;
n = 4;
a = [  0.47 - 0.34i, -0.40 + 0.54i,  0.60 + 0.01i,  0.80 - 1.02i;
-0.32 - 0.23i, -0.05 + 0.2i,  -0.26 - 0.44i, -0.43 + 0.17i;
0.35 - 0.60i, -0.52 - 0.34i,  0.87 - 0.11i, -0.34 - 0.09i;
0.89 + 0.71i, -0.45 - 0.45i, -0.02 - 0.57i,  1.14 - 0.78i;
-0.19 + 0.06i,  0.11 - 0.85i,  1.44 + 0.80i,  0.07 + 1.14i];

b = [ -1.08 - 2.59i,  2.22 + 2.35i;
-2.61 - 1.49i,  1.62 - 1.48i;
3.13 - 3.61i,  1.65 + 3.43i;
7.33 - 8.01i, -0.98 + 3.08i;
9.12 + 7.63i, -2.84 + 2.78i];

% QR factorization of A with column pivoting = Q*(R1 R2 )*(P^T)
%                                                (0  R22)
[qr, jpvt, tau, info] = f08bt( ...
a, zeros(n,1,'int64'));

% QRP'X = B, => RP'X = Q^HB = C; Compute C = Q^H B
[c, info] = f08au( ...
'Left', 'Conjugate Transpose', qr, tau, b);

% Determine the rank, k, of R relative to tol;
% Choose tol to reflect the relative accuracy of the input data
tol = 0.01;
k = find(abs(diag(qr)) <= tol*abs(qr(1,1)));
if numel(k) == 0
k = numel(diag(qr));
else
k = k(1)-1;
end

fprintf('\nTolerance used to estimate the rank of a\n     %11.2e\n', tol);
fprintf('Estimated rank of a\n        %d\n', k);

% Compute the RZ (TZ) factorization of the first k rows of (R1 R2)
[rz, taurz, info] = f08bv( ...
qr(1:k,:));

% Now, (TZ)P'X = C on first k rows of C
% Let ZP'X = T^{-1}C = Y (on first k rows)
y = zeros(n, 2);
y(1:k, :) = inv(triu(rz(1:k,1:k)))*c(1:k,:);

% ZP^T X = Y => P^T X = Z^H Y = W; Form W = Z^H Y.
[w, info] = f08bx( ...
'Left', 'ConjTrans', int64(n-k), rz, taurz, y);

% P^T X = W => X = PW,
x = zeros(n, 2);
for i=1:n
x(jpvt(i), :) = w(i, :);
end
fprintf('\nLeast-squares solution(s)\n');
disp(x);

% Compute estimates of the square roots of the residual sums of
% squares (2-norm of each of the columns of C2)
rnorm = [norm(c(k+1:m,1)), norm(c(k+1:m,2))];
fprintf('Square root(s) of the residual sum(s) of squares\n');
disp(rnorm);

f08bv example results

Tolerance used to estimate the rank of a
1.00e-02
Estimated rank of a
3

Least-squares solution(s)
1.1669 - 3.3224i  -0.5023 + 1.8323i
1.3486 + 5.5027i  -1.4418 - 1.6465i
4.1764 + 2.3435i   0.2908 + 1.4900i
0.6467 + 0.0107i  -0.2453 + 0.3951i

Square root(s) of the residual sum(s) of squares
0.2513    0.0810