- Interval Evaluation
- Interval Inclusions for Differential Equations
- Interval Methods for Linear Algebra
- Interval Inclusions for Equations
- Plot Intervals

Algorihms for interval solvers, and guaranteed inclusions.

functionieval(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)

functionievalder(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)

functionidgl(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)

functionidglder(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)

functionisimpson(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)

functionilgs(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)

functioniinv(A:interval)

Guaranteed interval inverse of the matrix A. See:

inv (Linear Algebra),

xinv (Exact Computation)

functionievalpoly(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)

functionipolyval(p:interval vector, t:interval)

Guaranteed evaluation of a polynomial p(t). See:

ievalpoly (Interval Solvers and Guaranteed Solutions)

functionibisect(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)

functioninewton(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)

functioninewton2(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)

functionplotintervals(r)

Adds plots of two dimensional intervals to a given plot. r is an nx2 vector of intervals. See:

mxmibisectfxy (Maxima Functions for Euler)