naginterfaces.library.linsys.real_​band_​solve

naginterfaces.library.linsys.real_band_solve(n, kl, ku, ab, b)

real_band_solve computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ band matrix, with $k_l$ subdiagonals and $k_u$ superdiagonals, and $X$ and $B$ are $n\times r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

For full information please refer to the NAG Library document for f04bb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f04/f04bbf.html

Parameters

nint

The number of linear equations $n$, i.e., the order of the matrix $A$.

klint

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

kuint

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

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

The $n\times n$ matrix $A$.

See Further Comments for further details.

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

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

Returns

abfloat, ndarray, shape $(2\times kl+ku+1,n)$

If no exception is raised, $ab$ is overwritten by details of the factorization.

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

If no exception is raised, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row $ipiv[i-1]$. $ipiv[i-1]=i$ indicates a row interchange was not required.

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

If the function exits successfully or $errno$ = $n$ + 1, the $n\times r$ solution matrix $X$.

rcondfloat

If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as $rcond=1/\left({\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\right)$.

errbndfloat

If the function exits successfully or $errno$ = $n$ + 1, an estimate of the forward error bound for a computed solution $\hat{x}$, such that ${\parallel \hat{x}-x\parallel }_1/{\parallel x\parallel }_1\le errbnd$, where $\hat{x}$ is a column of the computed solution returned in the array $b$ and $x$ is the corresponding column of the exact solution $X$. If $rcond$ is less than machine precision, $errbnd$ is returned as unity.

Raises

NagValueError

(errno $-6$)

On entry, $\text{ldab}=⟨value⟩$, $kl=⟨value⟩$ and $ku=⟨value⟩$.

Constraint: $\text{ldab}\ge 2\times kl+ku+1$.

(errno $-4$)

On entry, $\text{nrhs}=⟨value⟩$.

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

(errno $-3$)

On entry, $ku=⟨value⟩$.

Constraint: $ku\ge 0$.

(errno $-2$)

On entry, $kl=⟨value⟩$.

Constraint: $kl\ge 0$.

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $(i>0)\text{and}(i\le n)$)

Diagonal element $⟨value⟩$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

(errno $n+1$)

A solution has been computed, but $rcond$ is less than machine precision so that the matrix $A$ is numerically singular.

Notes

The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is the product of permutation matrices and unit lower triangular matrices with $k_l$ subdiagonals, and $U$ is upper triangular with $(k_l+k_u)$ superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

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

Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia