naginterfaces.library.eigen.complex_​triang_​svd

naginterfaces.library.eigen.complex_triang_svd(n, a, b, wantq, wantp)

complex_triang_svd returns all, or part, of the singular value decomposition of a complex upper triangular matrix.

For full information please refer to the NAG Library document for f02xu

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f02/f02xuf.html

Parameters

nint

$n$, the order of the matrix $R$.

If $n=0$, an immediate return is effected.

acomplex, array-like, shape $(n,n)$

The leading $n\times n$ upper triangular part of the array $a$ must contain the upper triangular matrix $R$.

bcomplex, array-like, shape $(:,\text{ncolb})$

Note: the required extent for this argument in dimension 1 is determined as follows: if $\text{ncolb}>0$: $n$; otherwise: $1$.

If $\text{ncolb}>0$, the leading $n\times \text{ncolb}$ part of the array $b$ must contain the matrix to be transformed.

wantqbool

Must be $True$ if the matrix $Q$ is required.

If $wantq=False$ then the array $q$ is not referenced.

wantpbool

Must be $True$ if the matrix $P^H$ is required, in which case $P^H$ is returned in the array $a$, otherwise $wantp$ must be $False$.

Returns

acomplex, ndarray, shape $(n,n)$

If $wantp=True$, the $n\times n$ part of $a$ will contain the $n\times n$ unitary matrix $P^H$, otherwise the $n\times n$ upper triangular part of $a$ is used as internal workspace, but the strictly lower triangular part of $a$ is not referenced.

bcomplex, ndarray, shape $(:,\text{ncolb})$

Is overwritten by the $n\times \text{ncolb}$ matrix $Q^{H}B$.

qcomplex, ndarray, shape $(:,:)$

If $wantq=True$, the leading $n\times n$ part of the array $q$ will contain the unitary matrix $Q$. Otherwise the array $q$ is not referenced.

svfloat, ndarray, shape $\left(n\right)$

The $n$ diagonal elements of the matrix $S$.

rworkfloat, ndarray, shape $(:)$

$rwork[n-1]$ contains the total number of iterations taken by the $QR$ algorithm.

The rest of the array is used as workspace.

Raises

NagValueError

(errno $-1$)

On entry, $\text{ldq}=⟨value⟩$ and $n=⟨value⟩$.

Constraint: if $wantq=True$, $\text{ldq}\ge n$.

(errno $-1$)

On entry, $\text{ldb}=⟨value⟩$, $n=⟨value⟩$ and $\text{ncolb}=⟨value⟩$.

Constraint: if $\text{ncolb}>0$, $\text{ldb}\ge n$.

(errno $-1$)

On entry, $\text{ncolb}=⟨value⟩$.

Constraint: $\text{ncolb}\ge 0$.

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

The $QR$ algorithm has failed to converge. $⟨value⟩$ singular values have not been found.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The $n\times n$ upper triangular matrix $R$ is factorized as

$$R=QS{P}^H\text{,}$$

where $Q$ and $P$ are $n\times n$ unitary matrices and $S$ is an $n\times n$ diagonal matrix with real non-negative diagonal elements, $s{v}_1,s{v}_2,\dots ,s{v}_n$, ordered such that

$$s{v}_1\ge s{v}_2\ge \cdots \ge s{v}_n\ge 0\text{.}$$

The columns of $Q$ are the left-hand singular vectors of $R$, the diagonal elements of $S$ are the singular values of $R$ and the columns of $P$ are the right-hand singular vectors of $R$.

Either or both of $Q$ and $P^H$ may be requested and the matrix $C$ given by

$$C=Q^{H}B\text{,}$$

where $B$ is an $n\times \text{ncolb}$ given matrix, may also be requested.

complex_triang_svd obtains the singular value decomposition by first reducing $R$ to bidiagonal form by means of Givens plane rotations and then using the $QR$ algorithm 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 if $K$ is any unitary diagonal matrix so that

$$K{K}^H=I\text{,}$$

then

$$A=\left(QK\right)S{\left(PK\right)}^H$$

is also a singular value decomposition of $A$.

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