NAG Library Function Document

nag_dgelsd (f08kcc)

void	nag_dgelsd (Nag_OrderType order, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, double s[], double rcond, Integer rank, NagError fail)

3 Description

nag_dgelsd (f08kcc) uses the singular value decomposition (SVD) of

A

, where

A

is a real

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 problem is solved in three steps:

1.	reduce the coefficient matrix $A$ to bidiagonal form with Householder transformations, reducing the original problem into a ‘bidiagonal least squares problem’ (BLS);
2.	solve the BLS using a divide-and-conquer approach;
3.	apply back all the Householder transformations to solve the original least squares problem.

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 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: 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

5: a[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

n

coefficient matrix

A

On exit: the contents of a are destroyed.

6: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

7: b[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, \max (1, m, n) \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

r

right-hand side matrix

B

On exit: b is overwritten by the

n

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.

8: pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, m, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

9: s[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array s must be at least

\max (1, \min (m, n))

On exit: the singular values of

A

in decreasing order.

10: rcond – doubleInput

On entry: used to determine the effective rank of

A

. Singular values

s [i - 1] \leq rcond \times s [0]

are treated as zero. If

rcond < 0

, machine precision is used instead.

11: rank – Integer *Output

On exit: the effective rank of

A

, i.e., the number of singular values which are greater than

rcond \times s [0]

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: The algorithm for computing the SVD failed to converge; $⟨value⟩$ off-diagonal elements of an intermediate bidiagonal form did not converge to zero.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $nrhs = ⟨value⟩$ .
Constraint: $nrhs \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
On entry, $pdb = ⟨value⟩$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pda \geq \max (1, m)$ .
On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq \max (1, n)$ .
On entry, $pdb = ⟨value⟩$ and $nrhs = ⟨value⟩$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
NE_INT_3: On entry, $pdb = ⟨value⟩$ , $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdb \geq \max (1, m, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

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

8 Parallelism and Performance

nag_dgelsd (f08kcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgelsd (f08kcc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this function is nag_zgelsd (f08kqc).

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} - 0.09 & - 1.56 & - 1.48 & - 1.09 & 0.08 & - 1.59 \\ 0.14 & 0.20 & - 0.43 & 0.84 & 0.55 & - 0.72 \\ - 0.46 & 0.29 & 0.89 & 0.77 & - 1.13 & 1.06 \\ 0.68 & 1.09 & - 0.71 & 2.11 & 0.14 & 1.24 \\ 1.29 & 0.51 & - 0.96 & - 1.27 & 1.74 & 0.34 \end{array}) and b = (\begin{array}{r} 7.4 \\ 4.3 \\ - 8.1 \\ 1.8 \\ 8.7 \end{array}) .

A tolerance of

0.01

is used to determine the effective rank of

A

NAG Library Function Documentnag_dgelsd (f08kcc)

+− 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 Library Function Document

nag_dgelsd (f08kcc)