NAG Library Function Document

1Purpose

nag_sparse_sym_precon_ichol_solve (f11jbc) solves a system of linear equations involving the incomplete Cholesky preconditioning matrix generated by nag_sparse_sym_chol_fac (f11jac).

2Specification

 #include #include
 void nag_sparse_sym_precon_ichol_solve (Integer n, const double a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipiv[], const Integer istr[], Nag_SparseSym_CheckData check, const double y[], double x[], NagError *fail)

3Description

nag_sparse_sym_precon_ichol_solve (f11jbc) solves a system of linear equations
 $Mx=y$
involving the preconditioning matrix $M=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, corresponding to an incomplete Cholesky decomposition of a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the f11 Chapter Introduction), as generated by nag_sparse_sym_chol_fac (f11jac).
In the above decomposition $L$ is a lower triangular sparse matrix with unit diagonal, $D$ is a diagonal matrix and $P$ is a permutation matrix. $L$ and $D$ are supplied to nag_sparse_sym_precon_ichol_solve (f11jbc) through the matrix
 $C=L+D-1-I$
which is a lower triangular n by n sparse matrix, stored in SCS format, as returned by nag_sparse_sym_chol_fac (f11jac). The permutation matrix $P$ is returned from nag_sparse_sym_chol_fac (f11jac) via the array ipiv.
It is envisaged that a common use of nag_sparse_sym_precon_ichol_solve (f11jbc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. nag_sparse_sym_precon_ichol_solve (f11jbc) is used for this purpose by the Black Box function nag_sparse_sym_chol_sol (f11jcc).
nag_sparse_sym_precon_ichol_solve (f11jbc) may also be used in combination with nag_sparse_sym_chol_fac (f11jac) to solve a sparse symmetric positive definite system of linear equations directly (see Section 9.4 in nag_sparse_sym_chol_fac (f11jac)). This use of nag_sparse_sym_precon_ichol_solve (f11jbc) is demonstrated in Section 10.
None.

5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $M$. This must be the same value as was supplied in the preceding call to nag_sparse_sym_chol_fac (f11jac).
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{a}\left[{\mathbf{la}}\right]$const doubleInput
On entry: the values returned in the array a by a previous call to nag_sparse_sym_chol_fac (f11jac).
3:    $\mathbf{la}$IntegerInput
On entry: the dimension of the arrays a, irow and icol. This must be the same value returned by the preceding call to nag_sparse_sym_chol_fac (f11jac).
4:    $\mathbf{irow}\left[{\mathbf{la}}\right]$const IntegerInput
5:    $\mathbf{icol}\left[{\mathbf{la}}\right]$const IntegerInput
6:    $\mathbf{ipiv}\left[{\mathbf{n}}\right]$const IntegerInput
7:    $\mathbf{istr}\left[{\mathbf{n}}+1\right]$const IntegerInput
On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_sym_chol_fac (f11jac).
8:    $\mathbf{check}$Nag_SparseSym_CheckDataInput
On entry: specifies whether or not the input data should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$
Checks are carried out on the values of n, irow, icol, ipiv and istr.
${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$
No checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ or $\mathrm{Nag_SparseSym_NoCheck}$.
9:    $\mathbf{y}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the right-hand side vector $y$.
10:  $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the solution vector $x$.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_ROWCOL_PIVOT
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).
NE_INVALID_SCS
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).
NE_INVALID_SCS_PRECOND
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).

7Accuracy

The computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεPLDLTPT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

8Parallelism and Performance

nag_sparse_sym_precon_ichol_solve (f11jbc) is not threaded in any implementation.

9.1Timing

The time taken for a call to nag_sparse_sym_precon_ichol_solve (f11jbc) is proportional to the value of nnzc returned from nag_sparse_sym_chol_fac (f11jac).

9.2Use of check

It is expected that a common use of nag_sparse_sym_precon_ichol_solve (f11jbc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. In this situation nag_sparse_sym_precon_ichol_solve (f11jbc) is likely to be called many times with the same matrix $M$. In the interests of both reliability and efficiency, you are recommended to set ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ for the first of such calls, and to set ${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$ for all subsequent calls.

10Example

This example reads in a symmetric positive definite sparse matrix $A$ and a vector $y$. It then calls nag_sparse_sym_chol_fac (f11jac), with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLTPT.$
Then it calls nag_sparse_sym_precon_ichol_solve (f11jbc) to solve the system
 $PLDLTPTx=y.$
It then repeats the exercise for the same matrix permuted with the bandwidth-reducing Reverse Cuthill–McKee permutation, calculated with nag_sparse_sym_rcm (f11yec).

10.1Program Text

Program Text (f11jbce.c)

10.2Program Data

Program Data (f11jbce.d)

10.3Program Results

Program Results (f11jbce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017