F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07CSF (ZGTTRS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07CSF (ZGTTRS) computes the solution to a complex system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$, where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices, using the $LU$ factorization returned by F07CRF (ZGTTRF).

## 2  Specification

 SUBROUTINE F07CSF ( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
 INTEGER N, NRHS, IPIV(*), LDB, INFO COMPLEX (KIND=nag_wp) DL(*), D(*), DU(*), DU2(*), B(LDB,*) CHARACTER(1) TRANS
The routine may be called by its LAPACK name zgttrs.

## 3  Description

F07CSF (ZGTTRS) should be preceded by a call to F07CRF (ZGTTRF), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
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. F07CSF (ZGTTRS) 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  Parameters

1:     TRANS – CHARACTER(1)Input
On entry: specifies the equations to be solved as follows:
${\mathbf{TRANS}}=\text{'N'}$
Solve $AX=B$ for $X$.
${\mathbf{TRANS}}=\text{'T'}$
Solve ${A}^{\mathrm{T}}X=B$ for $X$.
${\mathbf{TRANS}}=\text{'C'}$
Solve ${A}^{\mathrm{H}}X=B$ for $X$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     DL($*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array DL must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
5:     D($*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array D must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: must contain the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
6:     DU($*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array DU must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
7:     DU2($*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array DU2 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-2\right)$.
On entry: must contain the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
8:     IPIV($*$) – INTEGER arrayInput
Note: the dimension of the array IPIV must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
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 ${\mathbf{IPIV}}\left(i\right)$, and ${\mathbf{IPIV}}\left(i\right)$ must always be either $i$ or $\left(i+1\right)$, ${\mathbf{IPIV}}\left(i\right)=i$ indicating that a row interchange was not performed.
9:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
On exit: the $n$ by $r$ solution matrix $X$.
10:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07CSF (ZGTTRS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
11:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 =OεA1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x 1 x 1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, 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 routine F07CUF (ZGTCON) can be used to estimate the condition number of $A$ and F07CVF (ZGTRFS) can be used to obtain approximate error bounds.

The total number of floating point operations required to solve the equations $AX=B$ or ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$ is proportional to $nr$.
The real analogue of this routine is F07CEF (DGTTRS).

## 9  Example

This example solves the equations
 $AX=B ,$
where $A$ is the tridiagonal matrix
 $A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i$
and
 $B = 2.4-05.0i 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i .$

### 9.1  Program Text

Program Text (f07csfe.f90)

### 9.2  Program Data

Program Data (f07csfe.d)

### 9.3  Program Results

Program Results (f07csfe.r)