void	nag_zgttrs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer nrhs, const Complex dl[], const Complex d[], const Complex du[], const Complex du2[], const Integer ipiv[], Complex b[], Integer pdb, NagError *fail)

3 Description

nag_zgttrs (f07csc) should be preceded by a call to nag_zgttrf (f07crc), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

where

P

is a permutation matrix,

L

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

U

is an upper triangular band matrix, with two superdiagonals. nag_zgttrs (f07csc) then utilizes the factorization to solve the required equations.

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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $trans$ – Nag_TransTypeInput

On entry: specifies the equations to be solved as follows:

$trans = Nag_NoTrans$: Solve $A X = B$ for $X$ .
$trans = Nag_Trans$: Solve $A^{T} X = B$ for $X$ .
$trans = Nag_ConjTrans$: Solve $A^{H} X = B$ for $X$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

3: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

5: $dl [\dim]$ – const ComplexInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

6: $d [\dim]$ – const ComplexInput

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 upper triangular matrix

U

from the

L U

factorization of

A

7: $du [\dim]$ – const ComplexInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

elements of the first superdiagonal of

U

8: $du2 [\dim]$ – const ComplexInput

Note: the dimension, dim, of the array du2 must be at least

\max (1, n - 2)

On entry: must contain the

(n - 2)

elements of the second superdiagonal of

U

9: $ipiv [\dim]$ – const IntegerInput

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

\max (1, n)

On entry: must contain the

n

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]

, and

ipiv [i - 1]

must always be either

i

(i + 1)

ipiv [i - 1] = i

indicating that a row interchange was not performed.

10: $b [\dim]$ – ComplexInput/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: the

n

r

solution matrix

X

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

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error 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.
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: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.

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

Following the use of this function nag_zgtcon (f07cuc) can be used to estimate the condition number of

A

and nag_zgtrfs (f07cvc) can be used to obtain approximate error bounds.

8 Parallelism and Performance

nag_zgttrs (f07csc) is not threaded in any implementation.

9 Further Comments

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

A X = B

A^{T} X = B

A^{H} X = B

is proportional to

n r

The real analogue of this function is nag_dgttrs (f07cec).

10 Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array})

and

B = (\begin{array}{r} 2.4 - 5.0 i & 2.7 + 6.9 i \\ 3.4 + 18.2 i & - 6.9 - 5.3 i \\ - 14.7 + 9.7 i & - 6.0 - 0.6 i \\ 31.9 - 7.7 i & - 3.9 + 9.3 i \\ - 1.0 + 1.6 i & - 3.0 + 12.2 i \end{array}) .

NAG Library Function Documentnag_zgttrs (f07csc)

▸▿ 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_zgttrs (f07csc)