naginterfaces.library.blas.dtfsm

naginterfaces.library.blas.dtfsm(transr, side, uplo, trans, diag, alpha, a, b)

dtfsm performs one of the matrix-matrix operations

$$\begin{array}{c}\begin{array}{cc}B\leftarrow \alpha A^{-1}B\text{,}& B\leftarrow \alpha A^{-T}B\text{,}\\ B\leftarrow \alpha B{A}^{-1}\text{or}& B\leftarrow \alpha B{A}^{-T}\text{,}\end{array}\end{array}$$

where $A$ is a real triangular matrix stored in Rectangular Full Packed (RFP) format, $B$ is an $m\times n$ real matrix, and $\alpha$ is a real scalar. $A^{-T}$ denotes ${\left(A^T\right)}^{-1}$ or equivalently ${\left(A^{-1}\right)}^T$.

No test for singularity or near-singularity of $A$ is included in this function. Such tests must be performed before calling this function.

For full information please refer to the NAG Library document for f06wb

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f06/f06wbf.html

Parameters

transrstr, length 1

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

$transr=\text{'N'}$

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

$transr=\text{'T'}$

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

sidestr, length 1

Specifies whether $B$ is operated on from the left or the right, or similarly whether $A$ (or its transpose) appears to the left or right of the solution matrix in the linear system to be solved.

$side=\text{'L'}$

$B$ is pre-multiplied from the left. The system to be solved has the form $AX=\alpha B$ or $A^{T}X=\alpha B$.

$side=\text{'R'}$

$B$ is post-multiplied from the right. The system to be solved has the form $XA=\alpha B$ or $X{A}^T=\alpha B$.

uplostr, length 1

Specifies whether $A$ is upper or lower triangular.

$uplo=\text{'U'}$

$A$ is upper triangular.

$uplo=\text{'L'}$

$A$ is lower triangular.

transstr, length 1

Specifies whether the operation involves $A^{-1}$ or $A^{-T}$, i.e., whether or not $A$ is transposed in the linear system to be solved.

$trans=\text{'N'}$

The operation involves $A^{-1}$, i.e., $A$ is not transposed.

$trans=\text{'T'}$

The operation involves $A^{-T}$, i.e., $A$ is transposed.

diagstr, length 1

Specifies whether $A$ has nonunit or unit diagonal elements.

$diag=\text{'N'}$

The diagonal elements of $A$ are stored explicitly.

$diag=\text{'U'}$

The diagonal elements of $A$ are assumed to be $1$, the corresponding elements of $a$ are not referenced.

alphafloat

The scalar $\alpha$.

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

Note: the required length for this argument is determined as follows: if $side=\text{'L'}$: $m\times (m+1)/2$; if $side=\text{'R'}$: $n\times (n+1)/2$; otherwise: $0$.

$A$, the $m\times m$ triangular matrix $A$ if $side=\text{'L'}$ or the $n\times n$ triangular matrix $A$ if $side=\text{'R'}$, stored in RFP format (as specified by $transr$). The storage format is described in detail in the F07 Introduction. If $alpha=0.0$, $a$ is not referenced.

bfloat, array-like, shape $(m,n)$

The $m\times n$ matrix $B$.

If $alpha=0$, $b$ need not be set.

Returns

bfloat, ndarray, shape $(m,n)$

The updated matrix $B$, or similarly the solution matrix $X$.

Raises

NagValueError

(errno $1$)

On entry, error in parameter $transr$.

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

(errno $2$)

On entry, error in parameter $side$.

Constraint: $side=\text{'L'}$ or $\text{'R'}$.

(errno $3$)

On entry, error in parameter $uplo$.

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

(errno $4$)

On entry, error in parameter $trans$.

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

(errno $5$)

On entry, error in parameter $diag$.

Constraint: $diag=\text{'N'}$ or $\text{'U'}$.

(errno $6$)

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

Constraint: $m\ge 0$.

(errno $7$)

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

Constraint: $n\ge 0$.

Notes

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

dtfsm solves (for $X$) a triangular linear system of one of the forms

$$\begin{array}{c}\begin{array}{cc}AX=\alpha B\text{,}& A^{T}X=\alpha B\text{,}\\ XA=\alpha B\text{or}& X{A}^T=\alpha B\text{,}\end{array}\end{array}$$

where $A$ is a real triangular matrix stored in RFP format, $B$, $X$ are $m\times n$ real matrices, and $\alpha$ is a real scalar. The RFP storage format is described in the F07 Introduction.

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)