naginterfaces.library.lapacklin.dgttrs

naginterfaces.library.lapacklin.dgttrs(trans, dl, d, du, du2, ipiv, b)

dgttrs computes the solution to a real system of linear equations $AX=B$ or $A^{T}X=B$, where $A$ is an $n\times n$ tridiagonal matrix and $X$ and $B$ are $n\times r$ matrices, using the $LU$ factorization returned by dgttrf().

For full information please refer to the NAG Library document for f07ce

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07cef.html

Parameters

transstr, length 1

Specifies the equations to be solved as follows:

$trans=\text{'N'}$

Solve $AX=B$ for $X$.

$trans=\text{'T'}$ or $\text{'C'}$

Solve $A^{T}X=B$ for $X$.

dlfloat, array-like, shape $(n-1)$

Must contain the $(n-1)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.

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

Must contain the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

dufloat, array-like, shape $(n-1)$

Must contain the $(n-1)$ elements of the first superdiagonal of $U$.

du2float, array-like, shape $(n-2)$

Must contain the $(n-2)$ elements of the second superdiagonal of $U$.

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

Must contain the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row $ipiv[i-1]$, and $ipiv[i-1]$ must always be either $i$ or $(i+1)$, $ipiv[i-1]=i$ indicating that a row interchange was not performed.

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

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

Returns

bfloat, 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 $\text{nrhs}$.

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

Notes

dgttrs should be preceded by a call to dgttrf(), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as

$$A=PLU\text{,}$$

where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals. dgttrs then utilizes the factorization to solve the required equations.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug