NAG CL Interfacef11mfc (direct_​real_​gen_​solve)

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1Purpose

f11mfc solves a real sparse system of linear equations with multiple right-hand sides given an $LU$ factorization of the sparse matrix computed by f11mec.

2Specification

 #include
 void f11mfc (Nag_OrderType order, Nag_TransType trans, Integer n, const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], Integer nrhs, double b[], Integer pdb, NagError *fail)
The function may be called by the names: f11mfc, nag_sparse_direct_real_gen_solve or nag_superlu_solve_lu.

3Description

f11mfc solves a real system of linear equations with multiple right-hand sides $AX=B$ or ${A}^{\mathrm{T}}X=B$, according to the value of the argument trans, where the matrix factorization ${P}_{r}A{P}_{c}=LU$ corresponds to an $LU$ decomposition of a sparse matrix stored in compressed column (Harwell–Boeing) format, as computed by f11mec.
In the above decomposition $L$ is a lower triangular sparse matrix with unit diagonal elements and $U$ is an upper triangular sparse matrix; ${P}_{r}$ and ${P}_{c}$ are permutation matrices.

None.

5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{trans}$Nag_TransType Input
On entry: specifies whether $AX=B$ or ${A}^{\mathrm{T}}X=B$ is solved.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$AX=B$ is solved.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${A}^{\mathrm{T}}X=B$ is solved.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{iprm}\left[7×{\mathbf{n}}\right]$const Integer Input
On entry: the column permutation which defines ${P}_{c}$, the row permutation which defines ${P}_{r}$, plus associated data structures as computed by f11mec.
5: $\mathbf{il}\left[\mathit{dim}\right]$const Integer Input
Note: the dimension, dim, of the array il must be at least as large as the dimension of the array of the same name in f11mec.
On entry: records the sparsity pattern of matrix $L$ as computed by f11mec.
6: $\mathbf{lval}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array lval must be at least as large as the dimension of the array of the same name in f11mec.
On entry: records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by f11mec.
7: $\mathbf{iu}\left[\mathit{dim}\right]$const Integer Input
Note: the dimension, dim, of the array iu must be at least as large as the dimension of the array of the same name in f11mec.
On entry: records the sparsity pattern of matrix $U$ as computed by f11mec.
8: $\mathbf{uval}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array uval must be at least as large as the dimension of the array of the same name in f11mec.
On entry: records some nonzero values of matrix $U$ as computed by f11mec.
9: $\mathbf{nrhs}$Integer Input
On entry: $\mathit{nrhs}$, the number of right-hand sides in $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
10: $\mathbf{b}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the ${\mathbf{n}}×{\mathbf{nrhs}}$ right-hand side matrix $B$.
On exit: the ${\mathbf{n}}×{\mathbf{nrhs}}$ solution matrix $X$.
11: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
12: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_PERM_COL
Incorrect column permutations in array iprm.
NE_INVALID_PERM_ROW
Incorrect row permutations in array iprm.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $|E|≤c(n)ε|L||U|,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision, when partial pivoting is used.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$
where $\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$, and $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling f11mhc, and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling f11mgc.

8Parallelism and Performance

f11mfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11mfc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

f11mfc may be followed by a call to f11mhc to refine the solution and return an error estimate.

10Example

This example solves the system of equations $AX=B$, where
 $A=( 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 ) and B=( 1.56 3.12 -0.25 -0.50 3.60 7.20 1.33 2.66 0.52 1.04 ) .$
Here $A$ is nonsymmetric and must first be factorized by f11mec.

10.1Program Text

Program Text (f11mfce.c)

10.2Program Data

Program Data (f11mfce.d)

10.3Program Results

Program Results (f11mfce.r)