NAG Library Function Document

nag_zgttrs (f07csc)

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

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

Not applicable.

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)