f07th returns error bounds for the solution of a real triangular system of linear equations with multiple right-hand sides, AX=B or ATX=B.

Syntax

C#
public static void f07th(
	string uplo,
	string trans,
	string diag,
	int n,
	int nrhs,
	double[,] a,
	double[,] b,
	double[,] x,
	double[] ferr,
	double[] berr,
	out int info
)
Visual Basic
Public Shared Sub f07th ( _
	uplo As String, _
	trans As String, _
	diag As String, _
	n As Integer, _
	nrhs As Integer, _
	a As Double(,), _
	b As Double(,), _
	x As Double(,), _
	ferr As Double(), _
	berr As Double(), _
	<OutAttribute> ByRef info As Integer _
)
Visual C++
public:
static void f07th(
	String^ uplo, 
	String^ trans, 
	String^ diag, 
	int n, 
	int nrhs, 
	array<double,2>^ a, 
	array<double,2>^ b, 
	array<double,2>^ x, 
	array<double>^ ferr, 
	array<double>^ berr, 
	[OutAttribute] int% info
)
F#
static member f07th : 
        uplo : string * 
        trans : string * 
        diag : string * 
        n : int * 
        nrhs : int * 
        a : float[,] * 
        b : float[,] * 
        x : float[,] * 
        ferr : float[] * 
        berr : float[] * 
        info : int byref -> unit 

Parameters

uplo
Type: System..::..String
On entry: specifies whether A is upper or lower triangular.
uplo="U"
A is upper triangular.
uplo="L"
A is lower triangular.
Constraint: uplo="U" or "L".
trans
Type: System..::..String
On entry: indicates the form of the equations.
trans="N"
The equations are of the form AX=B.
trans="T" or "C"
The equations are of the form ATX=B.
Constraint: trans="N", "T" or "C".
diag
Type: System..::..String
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag="N"
A is a nonunit triangular matrix.
diag="U"
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag="N" or "U".
n
Type: System..::..Int32
On entry: n, the order of the matrix A.
Constraint: n0.
nrhs
Type: System..::..Int32
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
a
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: dim1max1,n
Note: the second dimension of the array a must be at least max1,n.
On entry: the n by n triangular matrix A.
  • If uplo="U", A is upper triangular and the elements of the array below the diagonal are not referenced.
  • If uplo="L", A is lower triangular and the elements of the array above the diagonal are not referenced.
  • If diag="U", the diagonal elements of A are assumed to be 1, and are not referenced.
b
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: dim1max1,n
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r right-hand side matrix B.
x
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: dim1max1,n
Note: the second dimension of the array x must be at least max1,nrhs.
On entry: the n by r solution matrix X, as returned by f07te.
ferr
Type: array<System..::..Double>[]()[][]
An array of size [nrhs]
On exit: ferr[j-1] contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
berr
Type: array<System..::..Double>[]()[][]
An array of size [nrhs]
On exit: berr[j-1] contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
info
Type: System..::..Int32%
On exit: info=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

f07th returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular system of linear equations with multiple right-hand sides AX=B or ATX=B. The method handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of f07th in terms of a single right-hand side b and solution x.
Given a computed solution x, the method computes the component-wise backward error β. This is the size of the smallest relative perturbation in each element of A and b such that x is the exact solution of a perturbed system
A+δAx=b+δbδaijβaij  and  δbiβbi.
Then the method estimates a bound for the component-wise forward error in the computed solution, defined by:
maxixi-x^i/maxixi
where x^ is the true solution.
For details of the method, see the F07 class.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA, LDB, LDX) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
ifail=-9000
An error occured, see message report.
ifail=-6000
Invalid Parameters value
ifail=-4000
Invalid dimension for array value
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value
ifail=-6000
Invalid Parameters value

Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

Parallelism and Performance

None.

Further Comments

A call to f07th, for each right-hand side, involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately n2 floating-point operations.
The complex analogue of this method is (F07TVF not in this release).

Example

This example solves the system of equations AX=B and to compute forward and backward error bounds, where
A=4.300.000.000.00-3.96-4.870.000.000.400.31-8.020.00-0.270.07-5.950.12  and  B=-12.90-21.5016.7514.93-17.556.33-11.048.09.

Example program (C#): f07the.cs

Example program data: f07the.d

Example program results: f07the.r

See Also