NAG Library Function Document

nag_real_sym_posdef_packed_lin_solve (f04bec)

void	nag_real_sym_posdef_packed_lin_solve (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, double ap[], double b[], Integer pdb, double rcond, double errbnd, NagError *fail)

3 Description

The Cholesky factorization is used to factor

A

A = U^{T} U

, if

uplo = Nag_Upper

, or

A = L L^{T}

, if

uplo = Nag_Lower

, where

U

is an upper triangular matrix and

L

is a lower triangular matrix. The factored form of

A

is then used to solve the system of equations

A X = B

4 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 http://www.netlib.org/lapack/lug

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

5 Arguments

1: order – Nag_OrderTypeInput

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

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: uplo – Nag_UploTypeInput

On entry: if

uplo = Nag_Upper

, the upper triangle of the matrix

A

is stored.

uplo = Nag_Lower

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: n – IntegerInput

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

4: nrhs – IntegerInput

On entry: the number of right-hand sides

r

, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

5: ap[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array ap must be at least

\max (1, n \times (n + 1) / 2)

On entry: the

n

n

symmetric matrix

A

. The upper or lower triangular part of the symmetric matrix is packed column-wise in a linear array. The

j

th column of

A

is stored in the array ap as follows:

if $uplo = Nag_Upper$ , $ap [i + (j - 1) j / 2] = a_{i j}$ for $1 \leq i \leq j$ ;
if $uplo = Nag_Lower$ , $ap [i + (j - 1) (2 n - j) / 2] = a_{i j}$ for $j \leq i \leq n$ .

See Section 8 below for further details.

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, the factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

, in the same storage format as

A

6: b[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

matrix of right-hand sides

B

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, the

n

r

solution matrix

X

7: pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

8: rcond – double *Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

9: errbnd – double *Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution

\hat{x}

, such that

{‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq 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, then errbnd is returned as unity.

10: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .; On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .
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.
NE_POS_DEF: 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.
NE_RCOND: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. nag_real_sym_posdef_packed_lin_solve (f04bec) uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Further Comments

The packed storage scheme is illustrated by the following example when

n = 4

and

uplo = Nag_Upper

. Two-dimensional storage of the symmetric matrix

A

\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{22} & a_{23} & a_{24} \\ a_{33} & a_{34} \\ a_{44} \end{matrix} (a_{i j} = a_{j i})

Packed storage of the upper triangle of

A

ap = [\begin{matrix} a_{11}, & a_{12}, & a_{22}, & a_{13}, & a_{23}, & a_{33}, & a_{14}, & a_{24}, & a_{34}, & a_{44} \end{matrix}]

The total number of floating point operations required to solve the equations

A X = B

is proportional to

(\frac{1}{3} n^{3} + n^{2} r)

. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of nag_real_sym_posdef_packed_lin_solve (f04bec) is nag_herm_posdef_packed_lin_solve (f04cec).

9 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite matrix

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

NAG Library Function Documentnag_real_sym_posdef_packed_lin_solve (f04bec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_real_sym_posdef_packed_lin_solve (f04bec)