Integer, Intent (In)	::	m, n, nrhs, lda, ldb, lwork
Integer, Intent (Out)	::	rank, info
Real (Kind=nag_wp), Intent (In)	::	rcond
Real (Kind=nag_wp), Intent (Inout)	::	s(), rwork()
Complex (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,)
Complex (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include <nag.h>

void	f08knf_ (const Integer m, const Integer n, const Integer nrhs, Complex a[], const Integer lda, Complex b[], const Integer ldb, double s[], const double rcond, Integer rank, Complex work[], const Integer lwork, double rwork[], Integer *info)

The routine may be called by the names f08knf, nagf_lapackeig_zgelss or its LAPACK name zgelss.

3 Description

f08knf uses the singular value decomposition (SVD) of

A

, where

A

is an

m \times n

matrix which may be rank-deficient.

Several right-hand side vectors

b

and solution vectors

x

can be handled in a single call; they are stored as the columns of the

m \times r

right-hand side matrix

B

and the

n \times r

solution matrix

X

The effective rank of

A

is determined by treating as zero those singular values which are less than rcond times the largest singular value.

4 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 https://www.netlib.org/lapack/lug

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

5 Arguments

1: $m$ – Integer Input: On entry: $m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – Integer Input: On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .
3: $nrhs$ – Integer Input: On entry: $r$ , the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$ .

Constraint: $nrhs \geq 0$ .
4: $a (lda, *)$ – Complex (Kind=nag_wp) array Input/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $m \times n$ matrix $A$ .

On exit: the first $\min (m, n)$ rows of $A$ are overwritten with its right singular vectors, stored row-wise.
5: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f08knf is called.

Constraint: $lda \geq \max (1, m)$ .
6: $b (ldb, *)$ – Complex (Kind=nag_wp) array Input/Output: Note: the second dimension of the array b must be at least $\max (1, nrhs)$ .

On entry: the $m \times r$ right-hand side matrix $B$ .

On exit: b is overwritten by the $n \times r$ solution matrix $X$ . If $m \geq n$ and $rank = n$ , the residual sum of squares for the solution in the $i$ th column is given by the sum of squares of the modulus of elements $n + 1, \dots, m$ in that column.
7: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f08knf is called.

Constraint: $ldb \geq \max (1, m, n)$ .
8: $s (*)$ – Real (Kind=nag_wp) array Output: Note: the dimension of the array s must be at least $\max (1, \min (m, n))$ .

On exit: the singular values of $A$ in decreasing order.
9: $rcond$ – Real (Kind=nag_wp) Input: On entry: used to determine the effective rank of $A$ . Singular values $s (i) \leq rcond \times s (1)$ are treated as zero. If $rcond < 0$ , machine precision is used instead.
10: $rank$ – Integer Output: On exit: the effective rank of $A$ , i.e., the number of singular values which are greater than $rcond \times s (1)$ .
11: $work (\max (1, lwork))$ – Complex (Kind=nag_wp) array Workspace: On exit: if $info = 0$ , the real part of $work (1)$ contains the minimum value of lwork required for optimal performance.
12: $lwork$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which f08knf is called.
If $lwork = −1$ , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, lwork should generally be larger. Consider increasing lwork by at least $nb \times \min (m, n)$ , where $nb$ is the optimal block size.

Constraint: $lwork = −1$ or
$lwork \geq 1$ and $lwork \geq 2 \times \min (m, n) + \max (m, n, nrhs)$ .
13: $rwork (*)$ – Real (Kind=nag_wp) array Workspace: Note: the dimension of the array rwork must be at least $\max (1, 5 \times \min (m, n))$ .
14: $info$ – Integer Output: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6 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.

$info > 0$: The algorithm for computing the SVD failed to converge; $⟨ value ⟩$ off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

7 Accuracy

See Section 4.5 of Anderson et al. (1999) for details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08knf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08knf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The real analogue of this routine is f08kaf.

10 Example

This example solves the linear least squares problem

\min_{x} {‖ b - A x ‖}_{2}

for the solution,

x

, of minimum norm, where

A = (\begin{array}{r} 0.47 - 0.34 i & - 0.40 + 0.54 i & 0.60 + 0.01 i & 0.80 - 1.02 i \\ - 0.32 - 0.23 i & - 0.05 + 0.20 i & - 0.26 - 0.44 i & - 0.43 + 0.17 i \\ 0.35 - 0.60 i & - 0.52 - 0.34 i & 0.87 - 0.11 i & - 0.34 - 0.09 i \\ 0.89 + 0.71 i & - 0.45 - 0.45 i & - 0.02 - 0.57 i & 1.14 - 0.78 i \\ - 0.19 + 0.06 i & 0.11 - 0.85 i & 1.44 + 0.80 i & 0.07 + 1.14 i \end{array})

and

b = (\begin{array}{r} - 1.08 - 2.59 i \\ - 2.61 - 1.49 i \\ 3.13 - 3.61 i \\ 7.33 - 8.01 i \\ 9.12 + 7.63 i \end{array}) .

A tolerance of

0.01

is used to determine the effective rank of

A

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.

f08kn: FL CL CPP AD PY MB

NAG FL Interfacef08knf (zgelss)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08knf (zgelss)