NAG Library Function Document

nag_real_cholesky_solve_mult_rhs (f04agc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_real_cholesky_solve_mult_rhs (f04agc) calculates the approximate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides,

A X = B

, where

A

has been factorized by nag_real_cholesky (f03aec).

2 Specification

#include <nag.h>

#include <nagf04.h>

void	nag_real_cholesky_solve_mult_rhs (Integer n, Integer nrhs, double a[], Integer tda, double p[], const double b[], Integer tdb, double x[], Integer tdx, NagError *fail)

3 Description

To solve a set of real linear equations

A X = B

where

A

is symmetric positive definite, nag_real_cholesky_solve_mult_rhs (f04agc) must be preceded by a call to nag_real_cholesky (f03aec) which computes a Cholesky factorization of

A

A = {L L}^{T}

, where

L

is lower triangular. The columns

x

of the solution

X

are found by forward and backward substitution in

L y = b

and

L^{T} x = y

, where

b

is a column of the right-hand sides.

4 References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5 Arguments

1: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 1$ .
2: nrhs – IntegerInput: On entry: $r$ , the number of right-hand sides.
Constraint: $nrhs \geq 1$ .
3: a[ $n \times tda$ ] – doubleInput: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: the upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$ , and the sub-diagonal elements of its Cholesky factor $L$ , as returned by nag_real_cholesky (f03aec).
4: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
5: p[n] – doubleInput: On entry: the reciprocals of the diagonal elements of $L$ , as returned by nag_real_cholesky (f03aec).
6: b[ $n \times tdb$ ] – const doubleInput: Note: the $(i, j)$ th element of the matrix $B$ is stored in $b [(i - 1) \times tdb + j - 1]$ .

On entry: the $n$ by $r$ right-hand side matrix $B$ . See also Section 9.
7: tdb – IntegerInput: On entry: the stride separating matrix column elements in the array b.

Constraint: $tdb \geq nrhs$ .
8: x[ $n \times tdx$ ] – doubleOutput: Note: the $(i, j)$ th element of the matrix $X$ is stored in $x [(i - 1) \times tdx + j - 1]$ .

On exit: the $n$ by $r$ solution matrix $X$ . See also Section 9.
9: tdx – IntegerInput: On entry: the stride separating matrix column elements in the array x.

Constraint: $tdx \geq nrhs$ .
10: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tda = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tda \geq n$ .
On entry, $tdb = ⟨value⟩$ while $nrhs = ⟨value⟩$ . These arguments must satisfy $tdb \geq nrhs$ .
On entry, $tdx = ⟨value⟩$ while $nrhs = ⟨value⟩$ . These arguments must satisfy $tdx \geq nrhs$ .
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
On entry, $nrhs = ⟨value⟩$ .
Constraint: $nrhs \geq 1$ .

7 Accuracy

The accuracy of the computed solutions depends on the conditioning of the original matrix. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_real_cholesky_solve_mult_rhs (f04agc) is approximately proportional to

n^{2} r

The function may be called with the same actual array supplied for arguments b and x, in which case the solution vectors will overwrite the right-hand sides.

10 Example

This example solves the set of linear equations

A X = B

where

A = (\begin{matrix} 5 & 7 & 6 & 5 \\ 7 & 10 & 8 & 7 \\ 6 & 8 & 10 & 9 \\ 5 & 7 & 9 & 10 \end{matrix}) and B = (\begin{matrix} 23 \\ 32 \\ 33 \\ 31 \end{matrix}) .

NAG Library Function Documentnag_real_cholesky_solve_mult_rhs (f04agc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_real_cholesky_solve_mult_rhs (f04agc)