﻿ Euler Math Toolbox - Reference

# Interval Solvers and Guaranteed Solutions

## Content

Algorihms for interval solvers, and guaranteed inclusions.

## Interval Evaluation

```function ieval (f\$:string, x:interval scalar, n:integer=10)
```
```  Better evaluation of the expression in f for the interval x.

The interval is split into sub-intervals for more accuracy.

See:   mxmieval (Maxima Functions for Euler),   ievalder (Interval Solvers and Guaranteed Solutions)
```
```function ievalder (f\$:string, fd\$:string, xi:interval scalar, n:integer=10)
```
```  Better evaluation of the expression in f for the interval x.

The derivative is used to improve the interval accuracy.
The interval is split into sub-intervals for more accuracy.

See:   mxmieval (Maxima Functions for Euler)
```

## Interval Inclusions for Differential Equations

```function idgl (f\$:string, x:real vector, y0:interval scalar)
```
```  Guaranteed inclusion of y'=f(t,y0;...) at points t with y(t[1])=y0.

This is a quick inclusion for a differential equation, which avoids
the use of any Taylor series. The inclusion is not narrow, however.
The function uses a simple Euler method.

The result is an interval vector of values.

See:   mxmidgl (Maxima Functions for Euler),   idglder (Interval Solvers and Guaranteed Solutions)
```
```function idglder (f\$:string, fx\$:string, fy\$:string, x:real vector, ..
y0:interval scalar)
```
```  Guaranteed inclusion the solution of y'=f(t,y0;...) at t with
y(t[1])=y0.

This function needs the partial derivatives of f to x and y.

The result is an interval vector of values.

f, fx and fy are functions in f(x,y), or expressions of x and y.
Additional arguments are passed to the functions.

See:   mxmidgl (Maxima Functions for Euler)
```
```function isimpson (f\$:string, der\$:string, a:number, b:number, ..
n:index=50)
```
```  Interval Simpson integral of f from a to b.

This function uses the Simpson method and its error estimate to get
guaranteed inclusions of integrals.

f : expression (must map to arguments and work for intervals)
der : expression for fourth derivative (like f)
a,b : interval bounds
n : number of subintervals

See:   mxmisimpson (Maxima Functions for Euler)
```

## Interval Methods for Linear Algebra

```function ilgs (A:interval, b:interval, R="", steps=100)
```
```  Guaranteed interval inclusion for the solution of A.x=b.

This function uses an interval agorithm, and an exact residuum
calculation. If the algorithm succeeds, the result is a guaranteed
inclusion for the solution of the linear system. Note that the
algorithm can only work for regular A, or interval matrices not
containing singular A.

A and b may be of interval or real type.

The optional R is a provided pseudo inverse to A.

See:   xlgs (Exact Computation)
```
```function iinv (A:interval)
```
```  Guaranteed interval inverse of the matrix A.

See:   inv (Linear Algebra),   xinv (Exact Computation)
```
```function ievalpoly (t:interval, p:interval vector)
```
```  Guaranteed evaluation of a polynomial p(t).

p contains the coefficients of a polynomial. Euler stores
polynomials starting with the constant coefficient.

See:   polyval (Euler Core),   xpolyval (Exact Computation)
```
```function ipolyval (p:interval vector, t:interval)
```
```  Guaranteed evaluation of a polynomial p(t).

See:   ievalpoly (Interval Solvers and Guaranteed Solutions)
```

## Interval Inclusions for Equations

```function ibisect (f:string, a:scalar, b:scalar=none, y:scalar=0)
```
```  Interval bisection algorithm to solve f(x)=y

See:   bisect (Numerical Algorithms),   inewton (Interval Solvers and Guaranteed Solutions)
```
```function inewton (f\$:string, df\$:string , xi: interval, yi:real scalar="", y=0)
```
```  Guaranteed interval inclusion of the zero of f.

df must compute an inclusion of the derivative of f for intervals
x. f and df must be functions of one scalar variable, or
expressions in x. Additional parameters after the semicolon are
passed to both functions.

The initial interval x must already contain a zero. If x is a
point, and not an interval, the function tries to get an initial
interval with the usual Newton method.

Returns {x0,f}: the solution and a flag, if the solution is
guaranteed.

See:   inewton2 (Interval Solvers and Guaranteed Solutions),   mxminewton (Maxima Functions for Euler),   inewton2 (Interval Solvers and Guaranteed Solutions)
```
```function inewton2 (f\$:string, Df\$:string, x:interval, check:integer=false)
```
```  Guaranteed inclusion of the zero of f, a function of several parameters.

Works like newton2, starting from a interval vector x which already
contains a solution. If x is no interval, the function tries to
find such an interval.

f and Df must be a function of a row vector x, or an expression in
x. f must return a row vector, and Df the Jacobi matrix of f.

Returns {x,valid}.

If check is false, the result is not checked for a guaranteed
inclusion. In this case the return value of valid can be checked
to learn, if the inclusion is a guaranteed inclusion. If checked
is true valid=0 will throw an error exception.

See:   newton2 (Numerical Algorithms)
```

## Plot Intervals

```function plotintervals (r)
```
```  Adds plots of two dimensional intervals to a given plot.

r is an nx2 vector of intervals.

See:   mxmibisectfxy (Maxima Functions for Euler)
```

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