NAG Library Routine Document

F08KAF (DGELSS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08KAF (DGELSS) computes the minimum norm solution to a real linear least squares problem

\min_{x} {‖b - A x‖}_{2} .

2 Specification

SUBROUTINE F08KAF (

M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO)

INTEGER	M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), B(LDB,), S(*), RCOND, WORK(max(1,LWORK))

The routine may be called by its LAPACK name dgelss.

3 Description

F08KAF (DGELSS) uses the singular value decomposition (SVD) of

A

, where

A

is an

m

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

r

right-hand side matrix

B

and the

n

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 http://www.netlib.org/lapack/lug

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

5 Parameters

1: M – INTEGERInput: On entry: $m$ , the number of rows of the matrix $A$ .
Constraint: $M \geq 0$ .
2: N – INTEGERInput: On entry: $n$ , the number of columns of the matrix $A$ .
Constraint: $N \geq 0$ .
3: NRHS – INTEGERInput: 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, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .
On entry: the $m$ by $n$ matrix $A$ .

On exit: the first $\min (m, n)$ rows of $A$ are overwritten with its right singular vectors, stored row-wise.
5: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F08KAF (DGELSS) is called.
Constraint: $LDA \geq \max (1, M)$ .
6: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array B must be at least $\max (1, NRHS)$ .
On entry: the $m$ by $r$ right-hand side matrix $B$ .

On exit: B is overwritten by the $n$ by $r$ 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 elements $n + 1, \dots, m$ in that column.
7: LDB – INTEGERInput: On entry: the first dimension of the array B as declared in the (sub)program from which F08KAF (DGELSS) is called.
Constraint: $LDB \geq \max (1, M, N)$ .
8: S( $*$ ) – REAL (KIND=nag_wp) arrayOutput: 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 – INTEGEROutput: On exit: the effective rank of $A$ , i.e., the number of singular values which are greater than $RCOND \times S (1)$ .
11: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , $WORK (1)$ contains the minimum value of LWORK required for optimal performance.
12: LWORK – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which F08KAF (DGELSS) 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: $LWORK \geq 1$ , and also
$LWORK \geq 3 \times \min (M, N) + \max (2 \times \min (M, N), \max (M, N), NRHS)$ .
13: INFO – INTEGEROutput: On exit: $INFO = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: 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; if $INFO = i$ , $i$ 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 Further Comments

The complex analogue of this routine is F08KNF (ZGELSS).

9 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} - 0.09 & 0.14 & - 0.46 & 0.68 & 1.29 \\ - 1.56 & 0.20 & 0.29 & 1.09 & 0.51 \\ - 1.48 & - 0.43 & 0.89 & - 0.71 & - 0.96 \\ - 1.09 & 0.84 & 0.77 & 2.11 & - 1.27 \\ 0.08 & 0.55 & - 1.13 & 0.14 & 1.74 \\ - 1.59 & - 0.72 & 1.06 & 1.24 & 0.34 \end{array}) and b = (\begin{array}{r} 7.4 \\ 4.2 \\ - 8.3 \\ 1.8 \\ 8.6 \\ 2.1 \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.

NAG Library Routine DocumentF08KAF (DGELSS)