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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_tridiag_solve (f04cc)

## Purpose

nag_linsys_complex_tridiag_solve (f04cc) computes the solution to a complex system of linear equations AX = B$AX=B$, where A$A$ is an n$n$ by n$n$ tridiagonal matrix and X$X$ and B$B$ are n$n$ by r$r$ matrices. An estimate of the condition number of A$A$ and an error bound for the computed solution are also returned.

## Syntax

[dl, d, du, du2, ipiv, b, rcond, errbnd, ifail] = f04cc(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, du2, ipiv, b, rcond, errbnd, ifail] = nag_linsys_complex_tridiag_solve(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

The LU$LU$ decomposition with partial pivoting and row interchanges is used to factor A$A$ as A = PLU$A=PLU$, where P$P$ is a permutation matrix, L$L$ is unit lower triangular with at most one nonzero subdiagonal element, and U$U$ is an upper triangular band matrix with two superdiagonals. The factored form of A$A$ is then used to solve the system of equations AX = B$AX=B$.
Note that the equations ATX = B${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

## 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

## Parameters

### Compulsory Input Parameters

1:     dl( : $:$) – complex array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the matrix A$A$.
2:     d( : $:$) – complex array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the matrix A$A$.
3:     du( : $:$) – complex array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ superdiagonal elements of the matrix A$A$
4:     b(ldb, : $:$) – complex array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ matrix of right-hand sides B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The dimension of the array d.
The number of linear equations n$n$, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides r$r$, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldb

### Output Parameters

1:     dl( : $:$) – complex array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, dl stores the (n1)$\left(n-1\right)$ multipliers that define the matrix L$L$ from the LU$LU$ factorization of A$A$.
2:     d( : $:$) – complex array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, d stores the n$n$ diagonal elements of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
3:     du( : $:$) – complex array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, du stores the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
4:     du2(n2${\mathbf{n}}-2$) – complex array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, du2 returns the (n2)$\left(n-2\right)$ elements of the second superdiagonal of U$U$.
5:     ipiv(n) – int64int32nag_int array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix P$P$; at the i$i$th step row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$. ipiv(i)${\mathbf{ipiv}}\left(i\right)$ will always be either i$i$ or (i + 1)$\left(i+1\right)$; ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
6:     b(ldb, : $:$) – complex array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
7:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A$A$, computed as rcond = 1 / (A1A11)${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
8:     errbnd – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that x1 / x1errbnd${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and x$x$ is the corresponding column of the exact solution X$X$. If rcond is less than machine precision, then errbnd is returned as unity.
9:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail < 0andifail999${\mathbf{ifail}}<0 \text{and} {\mathbf{ifail}}\ne -999$
If ifail = i${\mathbf{ifail}}=-i$, the i$i$th argument had an illegal value.
ifail = 999${\mathbf{ifail}}=-999$
Allocation of memory failed. The complex allocatable memory required is 2 × n$2×{\mathbf{n}}$. In this case the factorization and the solution X$X$ have been computed, but rcond and errbnd have not been computed.
ifail > 0andifailN${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{N}}$
If ifail = i${\mathbf{ifail}}=i$, uii${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor U$U$ is exactly singular, so the solution could not be computed.
W ifail = N + 1${\mathbf{ifail}}={\mathbf{N}}+1$
rcond is less than machine precision, so that the matrix A$A$ is numerically singular. A solution to the equations AX = B$AX=B$ has nevertheless been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b, $(A+E) x^=b,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε$\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A$A$ with respect to the solution of the linear equations. nag_linsys_complex_tridiag_solve (f04cc) uses the approximation E1 = εA1${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating point operations required to solve the equations AX = B$AX=B$ is proportional to nr$nr$. 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 real analogue of nag_linsys_complex_tridiag_solve (f04cc) is nag_linsys_real_tridiag_solve (f04bc).

## Example

```function nag_linsys_complex_tridiag_solve_example
dl = [ 1 - 2i;
1 + 1i;
2 - 3i;
1 + 1i];
d = [ -1.3 + 1.3i;
-1.3 + 1.3i;
-1.3 + 3.3i;
-0.3 + 4.3i;
-3.3 + 1.3i];
du = [ 2 - 1i;
2 + 1i;
-1 + 1i;
1 - 1i];
b = [ 2.4 - 5i,  2.7 + 6.9i;
3.4 + 18.2i,  -6.9 - 5.3i;
-14.7 + 9.7i,  -6 - 0.6i;
31.9 - 7.7i,  -3.9 + 9.3i;
-1 + 1.6i,  -3 + 12.2i];
[dlOut, dOut, duOut, du2, ipiv, bOut, rcond, errbnd, ifail] = ...
nag_linsys_complex_tridiag_solve(dl, d, du, b)
```
```

dlOut =

-0.7800 - 0.2600i
0.1620 - 0.4860i
-0.0452 - 0.0010i
-0.3979 - 0.0562i

dOut =

1.0000 - 2.0000i
1.0000 + 1.0000i
2.0000 - 3.0000i
1.0000 + 1.0000i
-1.3399 + 0.2875i

duOut =

-1.3000 + 1.3000i
-1.3000 + 3.3000i
-0.3000 + 4.3000i
-3.3000 + 1.3000i

du2 =

2.0000 + 1.0000i
-1.0000 + 1.0000i
1.0000 - 1.0000i

ipiv =

2
3
4
5
5

bOut =

1.0000 + 1.0000i   2.0000 - 1.0000i
3.0000 - 1.0000i   1.0000 + 2.0000i
4.0000 + 5.0000i  -1.0000 + 1.0000i
-1.0000 - 2.0000i   2.0000 + 1.0000i
1.0000 - 1.0000i   2.0000 - 2.0000i

rcond =

0.0054

errbnd =

2.0425e-14

ifail =

0

```
```function f04cc_example
dl = [ 1 - 2i;
1 + 1i;
2 - 3i;
1 + 1i];
d = [ -1.3 + 1.3i;
-1.3 + 1.3i;
-1.3 + 3.3i;
-0.3 + 4.3i;
-3.3 + 1.3i];
du = [ 2 - 1i;
2 + 1i;
-1 + 1i;
1 - 1i];
b = [ 2.4 - 5i,  2.7 + 6.9i;
3.4 + 18.2i,  -6.9 - 5.3i;
-14.7 + 9.7i,  -6 - 0.6i;
31.9 - 7.7i,  -3.9 + 9.3i;
-1 + 1.6i,  -3 + 12.2i];
[dlOut, dOut, duOut, du2, ipiv, bOut, rcond, errbnd, ifail] = f04cc(dl, d, du, b)
```
```

dlOut =

-0.7800 - 0.2600i
0.1620 - 0.4860i
-0.0452 - 0.0010i
-0.3979 - 0.0562i

dOut =

1.0000 - 2.0000i
1.0000 + 1.0000i
2.0000 - 3.0000i
1.0000 + 1.0000i
-1.3399 + 0.2875i

duOut =

-1.3000 + 1.3000i
-1.3000 + 3.3000i
-0.3000 + 4.3000i
-3.3000 + 1.3000i

du2 =

2.0000 + 1.0000i
-1.0000 + 1.0000i
1.0000 - 1.0000i

ipiv =

2
3
4
5
5

bOut =

1.0000 + 1.0000i   2.0000 - 1.0000i
3.0000 - 1.0000i   1.0000 + 2.0000i
4.0000 + 5.0000i  -1.0000 + 1.0000i
-1.0000 - 2.0000i   2.0000 + 1.0000i
1.0000 - 1.0000i   2.0000 - 2.0000i

rcond =

0.0054

errbnd =

2.0425e-14

ifail =

0

```