naginterfaces.library.lapacklin.zgbtrs

naginterfaces.library.lapacklin.zgbtrs(trans, kl, ku, ab, ipiv, b)

zgbtrs solves a complex band system of linear equations with multiple right-hand sides,

$$AX=B\text{,}{A}^{T}X=B\text{or}{A}^{H}X=B\text{,}$$

where $A$ has been factorized by zgbtrf().

For full information please refer to the NAG Library document for f07bs

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07bsf.html

Parameters

transstr, length 1

Indicates the form of the equations.

$trans=\text{'N'}$

$AX=B$ is solved for $X$.

$trans=\text{'T'}$

$A^{T}X=B$ is solved for $X$.

$trans=\text{'C'}$

$A^{H}X=B$ is solved for $X$.

klint

$k_l$, the number of subdiagonals within the band of the matrix $A$.

kuint

$k_u$, the number of superdiagonals within the band of the matrix $A$.

abcomplex, array-like, shape $(2\times kl+ku+1,n)$

The $LU$ factorization of $A$, as returned by zgbtrf().

ipivint, array-like, shape $\left(n\right)$

The pivot indices, as returned by zgbtrf().

bcomplex, array-like, shape $(n,\text{nrhs})$

The $n\times r$ right-hand side matrix $B$.

Returns

bcomplex, ndarray, shape $(n,\text{nrhs})$

The $n\times r$ solution matrix $X$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $trans$.

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

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-4$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

(errno $-5$)

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

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

Notes

zgbtrs is used to solve a complex band system of linear equations $AX=B$, $A^{T}X=B$ or $A^{H}X=B$, the function must be preceded by a call to zgbtrf() which computes the $LU$ factorization of $A$ as $A=PLU$. The solution is computed by forward and backward substitution.

If $trans=\text{'N'}$, the solution is computed by solving $PLY=B$ and then $UX=Y$.

If $trans=\text{'T'}$, the solution is computed by solving $U^{T}Y=B$ and then $L^T{P}^{T}X=Y$.

If $trans=\text{'C'}$, the solution is computed by solving $U^{H}Y=B$ and then $L^H{P}^{T}X=Y$.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore