naginterfaces.library.linsys.real_​tridiag_​solve

naginterfaces.library.linsys.real_tridiag_solve(n, dl, d, du, b)

real_tridiag_solve computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ tridiagonal matrix 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 f04bc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f04/f04bcf.html

Parameters

nint

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

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

Must contain the $(n-1)$ subdiagonal elements of the matrix $A$.

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

Must contain the $n$ diagonal elements of the matrix $A$.

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

Must contain the $(n-1)$ superdiagonal elements of the matrix $A$

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

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

Returns

dlfloat, ndarray, shape $(n-1)$

If no exception is raised, $dl$ is overwritten by the $(n-1)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.

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

If no exception is raised, $d$ is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

dufloat, ndarray, shape $(n-1)$

If no exception is raised, $du$ is overwritten by the $(n-1)$ elements of the first superdiagonal of $U$.

du2float, ndarray, shape $(n-2)$

If no exception is raised, $du2$ returns the $(n-2)$ elements of the second superdiagonal of $U$.

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 $ipiv[i-1]$. $ipiv[i-1]$ will always be either $i$ or $(i+1)$; $ipiv[i-1]=i$ 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 $-2$)

On entry, $\text{nrhs}=⟨value⟩$.

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

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

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

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 unit lower triangular with at most one nonzero subdiagonal element, and $U$ is an upper triangular band matrix with two superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

Note that the equations $A^{T}X=B$ may be solved by interchanging the order of the arguments $du$ and $dl$.

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