naginterfaces.library.blas.dsfrk

naginterfaces.library.blas.dsfrk(transr, uplo, trans, k, alpha, a, beta, c)

dsfrk performs one of the symmetric rank-$k$ update operations

$$C\leftarrow \alpha A{A}^T+\beta C\text{or}C\leftarrow \alpha A^{T}A+\beta C\text{,}$$

where $A$ is a real matrix, $C$ is an $n\times n$ real symmetric matrix stored in Rectangular Full Packed (RFP) format, and $\alpha$ and $\beta$ are real scalars.

For full information please refer to the NAG Library document for f06wc

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f06/f06wcf.html

Parameters

transrstr, length 1

Specifies whether the RFP representation of $C$ is normal or transposed.

$transr=\text{'N'}$

The matrix $C$ is stored in normal RFP format.

$transr=\text{'T'}$

The matrix $C$ is stored in transposed RFP format.

uplostr, length 1

Specifies whether the upper or lower triangular part of $C$ is stored in RFP format.

$uplo=\text{'U'}$

The upper triangular part of $C$ is stored in RFP format.

$uplo=\text{'L'}$

The lower triangular part of $C$ is stored in RFP format.

transstr, length 1

Specifies the operation to be performed.

$trans=\text{'N'}$

$C\leftarrow \alpha A{A}^T+\beta C$.

$trans=\text{'T'}$

$C\leftarrow \alpha A^{T}A+\beta C$.

kint

$k$, the number of columns of $A$ if $trans=\text{'N'}$, or the number of rows of $A$ if $trans=\text{'T'}$.

alphafloat

The scalar $\alpha$.

afloat, array-like, shape $(:,:)$

Note: the required extent for this argument in dimension 1 is determined as follows: if $trans=\text{'N'}$: $n$; if $trans=\text{'T'}$: $k$; otherwise: $0$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $trans=\text{'N'}$: $k$; if $trans=\text{'T'}$: $n$; otherwise: $0$.

The matrix $A$; $A$ is $n\times k$ if $trans=\text{'N'}$, or $k\times n$ if $trans=\text{'T'}$. If $alpha=0.0$, $a$ is not referenced.

betafloat

The scalar $\beta$.

cfloat, array-like, shape $(n\times (n+1)/2)$

The upper or lower triangular part (as specified by $uplo$) of the $n\times n$ symmetric matrix $C$, stored in RFP format (as specified by $transr$). The storage format is described in detail in the F07 Introduction.

Returns

cfloat, ndarray, shape $(n\times (n+1)/2)$

The updated matrix $C$, that is its upper or lower triangular part stored in RFP format.

Raises

NagValueError

(errno $1$)

On entry, error in parameter $transr$.

Constraint: $transr=\text{'N'}$ or $\text{'T'}$.

(errno $2$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $3$)

On entry, error in parameter $trans$.

Constraint: $trans=\text{'N'}$ or $\text{'T'}$.

(errno $4$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $5$)

On entry, error in parameter $k$.

Constraint: $k\ge 0$.

Notes

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

dsfrk performs one of the symmetric rank-$k$ update operations

$$C\leftarrow \alpha A{A}^T+\beta C\text{or}C\leftarrow \alpha A^{T}A+\beta C\text{,}$$

where $A$ is a real matrix, $C$ is an $n\times n$ real symmetric matrix stored in Rectangular Full Packed (RFP) format, and $\alpha$ and $\beta$ are real scalars. The RFP storage format is described in the F07 Introduction.

If $n=0$ or if $\beta =1.0$ and either $k=0$ or $\alpha =0.0$ then dsfrk returns immediately. If $\beta =0.0$ and either $k=0$ or $\alpha =0.0$ then $C$ is set to the zero matrix.

References

Gustavson, F G, Waśniewski, J, Dongarra, J J and Langou, J, 2010, Rectangular full packed format for Cholesky’s algorithm: factorization, solution, and inversion, ACM Trans. Math. Software (37, 2)