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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

```