NAG Library Function Document

nag_real_tridiag_lin_solve (f04bcc)

void	nag_real_tridiag_lin_solve (Nag_OrderType order, Integer n, Integer nrhs, double dl[], double d[], double du[], double du2[], Integer ipiv[], double b[], Integer pdb, double rcond, double errbnd, NagError *fail)

3 Description

The

L U

decomposition with partial pivoting and row interchanges is used to factor

A

A = P L U

, where

P

is a permutation matrix,

L

is unit lower triangular with at most one nonzero subdiagonal element, and

U

is an upper triangular band matrix with two superdiagonals. The factored form of

A

is then used to solve the system of equations

A X = B

Note that the equations

A^{T} X = B

may be solved by interchanging the order of the arguments du and dl.

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

Nag_ColMajor

2: n – IntegerInput

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: 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

4: dl[ $\dim$ ] – doubleInput/Output

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

subdiagonal elements of the matrix

A

On exit: if

fail . code =

NE_NOERROR, dl is overwritten by the

(n - 1)

multipliers that define the matrix

L

from the

L U

factorization of

A

5: d[ $\dim$ ] – doubleInput/Output

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

\max (1, n)

On entry: must contain the

n

diagonal elements of the matrix

A

On exit: if

fail . code =

NE_NOERROR, d is overwritten by the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

6: du[ $\dim$ ] – doubleInput/Output

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

superdiagonal elements of the matrix

A

On exit: if

fail . code =

NE_NOERROR, du is overwritten by the

(n - 1)

elements of the first superdiagonal of

U

7: du2[ $n - 2$ ] – doubleOutput

On exit: if

fail . code =

NE_NOERROR, du2 returns the

(n - 2)

elements of the second superdiagonal of

U

8: ipiv[n] – IntegerOutput

On exit: if

fail . code =

NE_NOERROR, the pivot indices that define the permutation matrix

P

; at the

i

th step row

i

of the matrix was interchanged with row

ipiv [i - 1]

ipiv [i - 1]

will always be either

i

(i + 1)

;

ipiv [i - 1] = i

indicates a row interchange was not required.

9: 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

10: 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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

11: rcond – double *Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

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

12: 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.

13: 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_RCOND: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
NE_SINGULAR: Diagonal element $⟨value⟩$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

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_tridiag_lin_solve (f04bcc) uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

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

8 Parallelism and Performance

nag_real_tridiag_lin_solve (f04bcc) is not threaded by NAG in any implementation.

nag_real_tridiag_lin_solve (f04bcc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

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

A X = B

is proportional to

n 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_tridiag_lin_solve (f04bcc) is nag_complex_tridiag_lin_solve (f04ccc).

10 Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{matrix} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{matrix}) and B = (\begin{matrix} 2.7 & 6.6 \\ - 0.5 & 10.8 \\ 2.6 & - 3.2 \\ 0.6 & - 11.2 \\ 2.7 & 19.1 \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_tridiag_lin_solve (f04bcc)

+− 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_tridiag_lin_solve (f04bcc)