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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgtrfs (f07ch)

## Purpose

nag_lapack_dgtrfs (f07ch) computes error bounds and refines the solution to a real system of linear equations AX = B $AX=B$ or ATX = B ${A}^{\mathrm{T}}X=B$, 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, using the LU $LU$ factorization returned by nag_lapack_dgttrf (f07cd) and an initial solution returned by nag_lapack_dgttrs (f07ce). Iterative refinement is used to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07ch(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dgtrfs(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgtrfs (f07ch) should normally be preceded by calls to nag_lapack_dgttrf (f07cd) and nag_lapack_dgttrs (f07ce). nag_lapack_dgttrf (f07cd) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix 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 in each column, and U $U$ is an upper triangular band matrix, with two superdiagonals. nag_lapack_dgttrs (f07ce) then utilizes the factorization to compute a solution, $\stackrel{^}{X}$, to the required equations. Letting $\stackrel{^}{x}$ denote a column of $\stackrel{^}{X}$, nag_lapack_dgtrfs (f07ch) computes a component-wise backward error, β $\beta$, the smallest relative perturbation in each element of A $A$ and b $b$ such that $\stackrel{^}{x}$ is the exact solution of a perturbed system
 (A + E) x̂ = b + f , with  |eij| ≤ β |aij| , and  |fj| ≤ β |bj| . $(A+E) x^=b+f , with |eij| ≤β |aij| , and |fj| ≤β |bj| .$
The function also estimates a bound for the component-wise forward error in the computed solution defined by max |xixi^| / max |xi^| $\mathrm{max}|{x}_{i}-\stackrel{^}{{x}_{i}}|/\mathrm{max}|\stackrel{^}{{x}_{i}}|$, where x $x$ is the corresponding column of the exact solution, X $X$.

## 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

## Parameters

### Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Specifies the equations to be solved as follows:
trans = 'N'${\mathbf{trans}}=\text{'N'}$
Solve AX = B$AX=B$ for X$X$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
Solve ATX = B${A}^{\mathrm{T}}X=B$ for X$X$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
2:     dl( : $:$) – double 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$A$.
3:     d( : $:$) – double 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$A$.
4:     du( : $:$) – double 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$A$.
5:     dlf( : $:$) – double array
Note: the dimension of the array dlf must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ multipliers that define the matrix L$L$ of the LU$LU$ factorization of A$A$.
6:     df( : $:$) – double array
Note: the dimension of the array df 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 upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
7:     duf( : $:$) – double array
Note: the dimension of the array duf must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
8:     du2( : $:$) – double array
Note: the dimension of the array du2 must be at least max (1,n2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
Must contain the (n2)$\left(n-2\right)$ elements of the second superdiagonal of U$U$.
9:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ 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)${\mathbf{ipiv}}\left(i\right)$, and ipiv(i)${\mathbf{ipiv}}\left(i\right)$ must always be either i$i$ or (i + 1)$\left(i+1\right)$, ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
10:   b(ldb, : $:$) – double 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$B$.
11:   x(ldx, : $:$) – double array
The first dimension of the array x 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$r$ initial solution matrix X$X$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays b, x The dimension of the arrays d, df, ipiv.
n$n$, 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 arrays b, x.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldb ldx work iwork

### Output Parameters

1:     x(ldx, : $:$) – double array
The first dimension of the array x 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ refined solution matrix X$X$.
2:     ferr(nrhs_p) – double array
Estimate of the forward error bound for each computed solution vector, such that jxj / jferr(j)${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{\stackrel{^}{x}}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$, where j${\stackrel{^}{x}}_{j}$ is the j$j$th column of the computed solution returned in the array x and xj${x}_{j}$ is the corresponding column of the exact solution X$X$. The estimate is almost always a slight overestimate of the true error.
3:     berr(nrhs_p) – double array
Estimate of the component-wise relative backward error of each computed solution vector j${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of A$A$ or B$B$ that makes j${\stackrel{^}{x}}_{j}$ an exact solution).
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: trans, 2: n, 3: nrhs_p, 4: dl, 5: d, 6: du, 7: dlf, 8: df, 9: duf, 10: du2, 11: ipiv, 12: b, 13: ldb, 14: x, 15: ldx, 16: ferr, 17: berr, 18: work, 19: iwork, 20: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## 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‖∞ = O(ε)‖A‖∞ $‖E‖∞=O(ε)‖A‖∞$
and ε $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖∞)/(‖x‖∞) ≤ κ(A) (‖E‖∞)/(‖A‖∞) , $‖ x^-x ‖∞ ‖x‖∞ ≤ κ(A) ‖E‖∞ ‖A‖∞ ,$
where κ(A) = A1 A $\kappa \left(A\right)={‖{A}^{-1}‖}_{\infty }{‖A‖}_{\infty }$, the condition number of A $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Function nag_lapack_dgtcon (f07cg) can be used to estimate the condition number of A $A$.

The total number of floating point operations required to solve the equations AX = B $AX=B$ or ATX = B ${A}^{\mathrm{T}}X=B$ is proportional to nr $nr$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this function is nag_lapack_zgtrfs (f07cv).

## Example

```function nag_lapack_dgtrfs_example
trans = 'No transpose';
dl = [3.4;
3.6;
7;
-6];
d = [3;
2.3;
-5;
-0.9;
7.1];
du = [2.1;
-1;
1.9;
8];
dlf = [0.8823529411764706;
0.01960784313725495;
0.1400560224089636;
-0.01479925303454714];
df = [3.4;
3.6;
7;
-6;
-1.015373482726424];
duf = [2.3;
-5;
-0.9;
7.1];
du2 = [-1;
1.9;
8];
ipiv = [int64(2);3;4;5;5];
b = [2.7, 6.6;
-0.5, 10.8;
2.6, -3.2;
0.6, -11.2;
2.7, 19.1];
x = [-4, 5;
7, -4;
3, -3;
-4, -2;
-3, 1];
[xOut, ferr, berr, info] = nag_lapack_dgtrfs(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x)
```
```

xOut =

-4     5
7    -4
3    -3
-4    -2
-3     1

ferr =

1.0e-13 *

0.0966
0.1286

berr =

1.0e-16 *

0.7401
0.4650

info =

0

```
```function f07ch_example
trans = 'No transpose';
dl = [3.4;
3.6;
7;
-6];
d = [3;
2.3;
-5;
-0.9;
7.1];
du = [2.1;
-1;
1.9;
8];
dlf = [0.8823529411764706;
0.01960784313725495;
0.1400560224089636;
-0.01479925303454714];
df = [3.4;
3.6;
7;
-6;
-1.015373482726424];
duf = [2.3;
-5;
-0.9;
7.1];
du2 = [-1;
1.9;
8];
ipiv = [int64(2);3;4;5;5];
b = [2.7, 6.6;
-0.5, 10.8;
2.6, -3.2;
0.6, -11.2;
2.7, 19.1];
x = [-4, 5;
7, -4;
3, -3;
-4, -2;
-3, 1];
[xOut, ferr, berr, info] = f07ch(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x)
```
```

xOut =

-4     5
7    -4
3    -3
-4    -2
-3     1

ferr =

1.0e-13 *

0.0966
0.1286

berr =

1.0e-16 *

0.7401
0.4650

info =

0

```