NAG Library Routine Document

F08BAF (DGELSY)

+− 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

F08BAF (DGELSY) computes the minimum norm solution to a real linear least squares problem

\min_{x} {‖b - A x‖}_{2}

using a complete orthogonal factorization of

A

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.

2 Specification

SUBROUTINE F08BAF (

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

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

The routine may be called by its LAPACK name dgelsy.

3 Description

The right-hand side vectors are stored as the columns of the

m

r

matrix

B

and the solution vectors in the

n

r

matrix

X

F08BAF (DGELSY) first computes a

Q R

factorization with column pivoting

A P = Q (\begin{array}{r} R_{11} & R_{12} \\ 0 & R_{22} \end{array}),

with

R_{11}

defined as the largest leading sub-matrix whose estimated condition number is less than

1 / RCOND

. The order of

R_{11}

, RANK, is the effective rank of

A

Then,

R_{22}

is considered to be negligible, and

R_{12}

is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization

A P = Q (\begin{array}{r} T_{11} & 0 \\ 0 & 0 \end{array}) Z .

The minimum norm solution is then

X = P Z^{T} (\begin{matrix} T_{11}^{- 1} Q_{1}^{T} b \\ 0 \end{matrix})

where

Q_{1}

consists of the first RANK columns of

Q

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

n

matrix

A

On exit: A has been overwritten by details of its complete orthogonal factorization.

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08BAF (DGELSY) 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

r

right-hand side matrix

B

On exit: the

n

r

solution matrix

X

7: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08BAF (DGELSY) is called.

Constraint:

LDB \geq \max (1, M, N)

8: JPVT( $*$ ) – INTEGER arrayInput/Output

Note: the dimension of the array JPVT must be at least

\max (1, N)

On entry: if

JPVT (i) \neq 0

, the

i

th column of

A

is permuted to the front of

A P

, otherwise column

i

is a free column.

On exit: if

JPVT (i) = k

, then the

i

th column of

A P

was the

k

th column of

A

9: RCOND – REAL (KIND=nag_wp)Input

On entry: used to determine the effective rank of

A

, which is defined as the order of the largest leading triangular sub-matrix

R_{11}

in the

Q R

factorization of

A

, whose estimated condition number is

< 1 / RCOND

Suggested value: if the condition number of A is not known then

RCOND = \sqrt{(ε) / 2}

(where

ε

is machine precision, see X02AJF) is a good choice. Negative values or values less than machine precision should be avoided since this will cause A to have an effective

rank = \min (M, N)

that could be larger than its actual rank, leading to meaningless results.

10: RANK – INTEGEROutput

On exit: the effective rank of

A

, i.e., the order of the sub-matrix

R_{11}

. This is the same as the order of the sub-matrix

T_{11}

in the complete orthogonal factorization of

A

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 F08BAF (DGELSY) is called.

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 \geq \max (k + 2 \times N + nb \times (N + 1), 2 \times k + nb \times NRHS),

where

k = \min (M, N)

and

nb

is the optimal block size.

Constraint:

LWORK \geq k + \max (2 \times k, N + 1, k + NRHS), where ​ k = \min (M, N)

LWORK = - 1

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.

7 Accuracy

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

8 Further Comments

The complex analogue of this routine is F08BNF (ZGELSY).

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 DocumentF08BAF (DGELSY)

+− 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

NAG Library Routine Document

F08BAF (DGELSY)