NAG Library Function Document

nag_real_svd (f02wec)

void	nag_real_svd (Integer m, Integer n, double a[], Integer tda, Integer ncolb, double b[], Integer tdb, Nag_Boolean wantq, double q[], Integer tdq, double sv[], Nag_Boolean wantp, double pt[], Integer tdpt, Integer iter, double e[], Integer failinfo, NagError *fail)

3 Description

The

m

n

matrix

A

is factorized as

A = {Q D P}^{T}

where

\begin{matrix} D = (\begin{matrix} S \\ 0 \end{matrix}) & m > n \\ D = S, & m = n \\ D = (\begin{matrix} S & 0 \end{matrix}) & m < n \end{matrix} .

Q

is an

m

m

orthogonal matrix,

P

is an

n

n

orthogonal matrix and

S

is a

\min (m, n)

\min (m, n)

diagonal matrix with non-negative diagonal elements,

{s v}_{1}, {s v}_{2}, \dots, {s v}_{\min (m, n)}

, ordered such that

{s v}_{1} \geq {s v}_{2} \geq \dots \geq {s v}_{\min (m, n)} \geq 0 .

The first

\min (m, n)

columns of

Q

are the left-hand singular vectors of

A

, the diagonal elements of

S

are the singular values of

A

and the first

\min (m, n)

columns of

P

are the right-hand singular vectors of

A

Either or both of the left-hand and right-hand singular vectors of

A

may be requested and the matrix

C

given by

C = Q^{T} B

where

B

is an

m

ncolb

given matrix, may also be requested.

The function obtains the singular value decomposition by first reducing

A

to upper triangular form by means of Householder transformations, from the left when

m \geq n

and from the right when

m < n

. The upper triangular form is then reduced to bidiagonal form by Givens plane rotations and finally the

Q R

algorithm is used to obtain the singular value decomposition of the bidiagonal form.

Good background descriptions to the singular value decomposition are given in Dongarra et al. (1979), Hammarling (1985) and Wilkinson (1978). Note that this function is not based on the LINPACK routine SSVDC.

Note that if

K

is any orthogonal diagonal matrix such that

{K K}^{T} = I,

so that

K

has elements

+ 1

- 1

on the diagonal, then

A = (\begin{matrix} Q & K \end{matrix}) D {(\begin{matrix} P & K \end{matrix})}^{T}

is also a singular value decomposition of

A

4 References

Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979) LINPACK Users' Guide SIAM, Philadelphia

Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25

Wilkinson J H (1978) Singular Value Decomposition – Basic Aspects Numerical Software – Needs and Availability (ed D A H Jacobs) Academic Press

5 Arguments

1: m – IntegerInput: On entry: the number of rows, $m$ , of the matrix $A$ .
Constraint: $m \geq 0$ .
When $m = 0$ then an immediate return is effected.
2: n – IntegerInput: On entry: the number of columns, $n$ , of the matrix $A$ .
Constraint: $n \geq 0$ .
When $n = 0$ then an immediate return is effected.
3: a[ $m \times tda$ ] – doubleInput/Output: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: the leading $m$ by $n$ part of the array a must contain the matrix $A$ whose singular value decomposition is required.

On exit: if $m \geq n$ and $wantq = Nag_TRUE$ , then the leading $m$ by $n$ part of a will contain the first $n$ columns of the orthogonal matrix $Q$ .
If $m < n$ and $wantp = Nag_TRUE$ , then the leading $m$ by $n$ part of a will contain the first $m$ rows of the orthogonal matrix $P^{T}$ .

If $m \geq n$ and $wantq = Nag_FALSE$ and $wantp = Nag_TRUE$ , then the leading $n$ by $n$ part of a will contain the first $n$ rows of the orthogonal matrix $P^{T}$ .

Otherwise the contents of the leading $m$ by $n$ part of a are indeterminate.
4: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
5: ncolb – IntegerInput: On entry: $ncolb$ , the number of columns of the matrix $B$ . When $ncolb = 0$ the array b is not referenced and may be NULL.
Constraint: $ncolb \geq 0$ .
6: b[ $m \times tdb$ ] – doubleInput/Output: Note: the $(i, j)$ th element of the matrix $B$ is stored in $b [(i - 1) \times tdb + j - 1]$ .

On entry: if $ncolb > 0$ , the leading $m$ by $ncolb$ part of the array b must contain the matrix to be transformed. If $ncolb = 0$ the array b is not referenced and may be NULL.

On exit: b is overwritten by the $m$ by $ncolb$ matrix $Q^{T} B$ .
7: tdb – IntegerInput: On entry: the stride separating matrix column elements in the array b.

Constraint: if $ncolb > 0$ then $tdb \geq ncolb$ .
8: wantq – Nag_BooleanInput: On entry: wantq must be Nag_TRUE, if the left-hand singular vectors are required. If wantq = Nag_FALSE, then the array q is not referenced and may be NULL.
9: q[ $m \times tdq$ ] – doubleOutput: Note: the $(i, j)$ th element of the matrix $Q$ is stored in $q [(i - 1) \times tdq + j - 1]$ .

On exit: if $m < n$ and $wantq = Nag_TRUE$ , the leading $m$ by $m$ part of the array q will contain the orthogonal matrix $Q$ . Otherwise the array q is not referenced and may be NULL.
10: tdq – IntegerInput: On entry: the stride separating matrix column elements in the array q.

Constraint: if $m < n$ and $wantq = Nag_TRUE$ , $tdq \geq m$
11: sv[ $\min (m, n)$ ] – doubleOutput: On exit: the $\min (m, n)$ diagonal elements of the matrix $S$ .
12: wantp – Nag_BooleanInput: On entry: wantp must be Nag_TRUE if the right-hand singular vectors are required. If $wantp = Nag_FALSE$ , then the array pt is not referenced and may be NULL.
13: pt[ $n \times tdpt$ ] – doubleOutput: Note: the $(i, j)$ th element of the matrix is stored in $pt [(i - 1) \times tdpt + j - 1]$ .

On exit: if $m \geq n$ and wantq and wantp are Nag_TRUE, the leading $n$ by $n$ part of the array pt will contain the orthogonal matrix $P^{T}$ . Otherwise the array pt is not referenced and may be NULL.
14: tdpt – IntegerInput: On entry: the stride separating matrix column elements in the array pt.

Constraint: if $m \geq n$ and $wantq = Nag_TRUE$ and $wantp = Nag_TRUE$ , $tdpt \geq n$
15: iter – Integer *Output: On exit: the total number of iterations taken by the $Q R$ algorithm.
16: e[ $\min (m, n)$ ] – doubleOutput: On exit: if the error NE_QR_NOT_CONV occurs the array e contains the super diagonal elements of matrix $E$ in the factorization of $A$ according to $A = {Q E P}^{T}$ . See Section 6 for further details.
17: failinfo – Integer *Output: On exit: if the error NE_QR_NOT_CONV occurs failinfo contains the number of singular values which may not have been found correctly. See Section 6 for details.
18: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tda = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tda \geq n$ .
On entry, $tdb = ⟨value⟩$ while $ncolb = ⟨value⟩$ . These arguments must satisfy $tdb \geq ncolb$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $ncolb = ⟨value⟩$ .
Constraint: $ncolb \geq 0$ .
NE_QR_NOT_CONV: The $Q R$ algorithm has failed to converge in $⟨value⟩$ iterations. Singular values 1,2, $\dots$ ,failinfo may not have been found correctly and the remaining singular values may not be the smallest. The matrix $A$ will nevertheless have been factorized as $A = {Q E P}^{T}$ , where the leading $\min (m, n)$ by $\min (m, n)$ part of $E$ is a bidiagonal matrix with $sv [0], sv [1], \dots, sv [\min (m, n - 1)]$ as the diagonal elements and $e [0], e [1], \dots, e [\min (m, n - 2)]$ as the superdiagonal elements. This failure is not likely to occur.
NE_TDP_LT_N: On entry, $tdpt = ⟨value⟩$ while $n = ⟨value⟩$ . When wantq and wantp are Nag_TRUE and $m \geq n$ then relationship $tdpt \geq n$ must be satisfied.
NE_TDQ_LT_M: On entry, $tdq = ⟨value⟩$ while $m = ⟨value⟩$ . When wantq is Nag_TRUE and $m < n$ then relationship $tdq \geq m$ must be satisfied.

7 Accuracy

The computed factors

Q

D

and

P

satisfy the relation

{Q D P}^{T} = A + E

where

‖E‖ \leq c ε ‖A‖

ε

being the machine precision,

c

is a modest function of

m

and

n

and

.

denotes the spectral (two) norm. Note that

‖A‖ = {s v}_{1}

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

For this function two examples are presented. There is a single example program for nag_real_svd (f02wec), with a main program and the code to solve the two example problems is given in the functions ex1 and ex2.

Example 1 (ex1)

To find the singular value decomposition of the 5 by 3 matrix

A = (\begin{matrix} 2.0 & 2.5 & 2.5 \\ 2.0 & 2.5 & 2.5 \\ 1.6 & - 0.4 & 2.8 \\ 2.0 & - 0.5 & 0.5 \\ 1.2 & - 0.3 & - 2.9 \end{matrix})

together with the vector

Q^{T} b

for the vector

b = (\begin{matrix} 1.1 \\ 0.9 \\ 0.6 \\ 0.0 \\ - 0.8 \end{matrix}) .

Example 2 (ex2)

To find the singular value decomposition of the 3 by 5 matrix

A = (\begin{matrix} 2.0 & 2.0 & 1.6 & 2.0 & 1.2 \\ 2.5 & 2.5 & - 0.4 & - 0.5 & - 0.3 \\ 2.5 & 2.5 & - 2.8 & 0.5 & - 2.9 \end{matrix}) .

NAG Library Function Documentnag_real_svd (f02wec)

+− 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_real_svd (f02wec)