naginterfaces.library.linsys.real_​posdef_​tridiag_​solve

naginterfaces.library.linsys.real_posdef_tridiag_solve(d, e, b)

real_posdef_tridiag_solve computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ symmetric positive definite 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 f04bg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f04/f04bgf.html

Parameters

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

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

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

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

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

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

Returns

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

If the function exits successfully or $errno$ = $n$ + 1, $d$ is overwritten by the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^T$ factorization of $A$.

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

If the function exits successfully or $errno$ = $n$ + 1, $e$ is overwritten by the $(n-1)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $LD{L}^T$ factorization of $A$. ($e$ can also be regarded as the superdiagonal of the unit upper bidiagonal factor $U$ from the $U^{T}DU$ factorization of $A$.)

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

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

rcondfloat

If the function exits successfully or $errno$ = $n$ + 1, 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: $n\ge 0$.

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

The principal minor of order $⟨value⟩$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.

Warns

NagAlgorithmicWarning

(errno $n+1$)

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

Notes

$A$ is factorized as $A=LD{L}^T$, where $L$ is a unit lower bidiagonal matrix and $D$ is diagonal, and the factored form of $A$ is then used to solve the system of 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

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