SVD of complex matrix

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NAG[f02xec] NAG[nag_complex_svd] - SVD of complex matrix

Calling Sequence

f02xec(a, wantq, sv, wantp, iter, e, failinfo, 'm'=m, 'n'=n, 'tda'=tda, 'ncolb'=ncolb, 'b'=b, 'tdb'=tdb, 'q'=q, 'tdq'=tdq, 'ph'=ph, 'tdph'=tdph, 'fail'=fail)

nag_complex_svd(. . .)

Parameters

a - Matrix(1..m, 1..tda, datatype=complex[8], order=C_order);

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

On exit: if and , then the leading by part of a will contain the first columns of the unitary matrix .

If and , then the leading by part of a will contain the first rows of the unitary matrix .

If and and , then the leading by part of a will contain the first rows of the unitary matrix .

Otherwise the contents of the leading by part of a are indeterminate.

wantq - boolean;

On entry: wantq must be true if the left-hand singular vectors are required. If then the array q is not referenced.

sv - Vector(1.., datatype=float[8]);

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

On exit: the diagonal elements of the matrix .

wantp - boolean;

On entry: wantp must be true if the right-hand singular vectors are required. If wantp = false then the array ph is not referenced.

iter - assignable;

Note: On exit the variable iter will have a value of type integer.

On exit: the total number of iterations taken by the algorithm.

e - Vector(1.., datatype=float[8]);

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

On exit: if the error NE_QR_NOT_CONV occurs the array e contains the super-diagonal elements of matrix in the factorization of according to . See Section [Error Indicators and Warnings] for further details.

failinfo - assignable;

Note: On exit the variable failinfo will have a value of type integer.

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 [Error Indicators and Warnings] for details.

'm'=m - integer; (optional)

Default value: the first dimension of the arrays a, b, q.

On entry: the number of rows, , of the matrix .

Constraint: ..

When then an immediate return is effected.

'n'=n - integer; (optional)

Default value: the first dimension of the array ph and the second dimension of the array phthe array a.

On entry: the number of columns, , of the matrix .

Constraint: ..

When then an immediate return is effected.

'tda'=tda - integer; (optional)

On entry: the second dimension of the array a as declared in the function from which nag_complex_svd (f02xec) is called.

Constraint: . .

'ncolb'=ncolb - integer; (optional)

Default value: the second dimension of the array b.

On entry: , the number of columns of the matrix . When the array b is not referenced.

Constraint: . .

'b'=b - Matrix(1..m, 1..tdb, datatype=complex[8], order=C_order); (optional)

On entry: if , the leading by part of the array b must contain the matrix to be transformed.

If the array b is not referenced and may be set to the null pointer, i.e., (Complex *)0.

On exit: b is overwritten by the by matrix .

'tdb'=tdb - integer; (optional)

On entry: the second dimension of the array b as declared in the function from which nag_complex_svd (f02xec) is called.

Constraint: if then . .

'q'=q - Matrix(1..m, 1..tdq, datatype=complex[8], order=C_order); (optional)

On exit: if and , the leading by part of the array q will contain the unitary matrix . Otherwise the array q is not referenced and may be set to the null pointer, i.e., (Complex *)0.

'tdq'=tdq - integer; (optional)

On entry: the second dimension of the array q as declared in the function from which nag_complex_svd (f02xec) is called.

Constraint: if and , . .

'ph'=ph - Matrix(1..n, 1..tdph, datatype=complex[8], order=C_order); (optional)

On exit: if and wantq and wantp are true, the leading by part of the array ph will contain the unitary matrix . Otherwise the array ph is not referenced and may be set to the null pointer, i.e., (Complex *)0.

'tdph'=tdph - integer; (optional)

On entry: the second dimension of the array ph as declared in the function from which nag_complex_svd (f02xec) is called.

Constraint: if and and , . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_complex_svd (f02xec) returns all, or part, of the singular value decomposition of a general complex matrix.

Description

The by matrix is factorized as

where

[[D=([[S],[0]]),m>n],[D=S ,m=n],[D=([[S,0]]),m<n]]

is an by unitary matrix, is an by unitary matrix and is a by diagonal matrix with real non-negative diagonal elements, , ordered such that

The first columns of are the left-hand singular vectors of , the diagonal elements of are the singular values of and the first columns of are the right-hand singular vectors of .

Either or both of the left-hand and right-hand singular vectors of may be requested and the matrix given by

where is an by given matrix, may also be requested.

The function obtains the singular value decomposition by first reducing to upper triangular form by means of Householder transformations, from the left when and from the right when The upper triangular form is then reduced to bidiagonal form by Givens plane rotations and finally the 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 CSVDC.

Note that if is any unitary diagonal matrix such that

then

is also a singular value decomposition of .

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_INT_ARG_LT"

On entry, m must not be less than 0: .

"NE_QR_NOT_CONV"

The QR algorithm has failed to converge in iterations. Singular values may not have been found correctly and the remaining singular values may not be the smallest. The matrix will nevertheless have been factorized as , where the leading by part of is a bidiagonal matrix with as the diagonal elements and as the super-diagonal elements. This failure is not likely to occur.

	Accuracy
	The computed factors , and satisfy the relation where , being the machine precision, is a modest function of and and denotes the spectral (two) norm. Note that .

	Further Comments
	None.

Examples

m := 5:
n := 3:
tda := 3:
ncolb := 1:
tdb := 1:
wantq := true:
tdq := 1:
wantp := true:
tdph := 3:
a := Matrix([[0 +0.5*I , -0.5 +1.5*I , -1 +1*I ], [0.4 +0.3*I , 0.9 +1.3*I , 0.2 +1.4*I ], [0.4 +0*I , -0.4 +0.4*I , 1.8 +0*I ], [0.3 -0.4*I , 0.1 +0.7*I , 0 +0*I ], [0 -0.3*I , 0.3 +0.3*I , 0 +2.4*I ]], datatype=complex[8], order='C_order'):
b := Matrix([[-0.55 +1.05*I ], [0.49 +0.93*I ], [0.5600000000000001 -0.16*I ], [0.39 +0.23*I ], [1.13 +0.83*I ]], datatype=complex[8], order='C_order'):
q := Matrix(0, 0, datatype=complex[8], order='C_order'):
sv := Vector(3, datatype=float[8]):
ph := Matrix(3, 3, datatype=complex[8], order='C_order'):
e := Vector(3, datatype=float[8]):
NAG:-f02xec(a, wantq, sv, wantp, iter, e, failinfo, 'm' = m, 'n' = n, 'tda' = tda, 'ncolb' = ncolb, 'b' = b, 'tdb' = tdb, 'q' = q, 'tdq' = tdq, 'ph' = ph, 'tdph' = tdph):