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Interval Solvers and Guaranteed Solutions

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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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