% This notebook introduces you into interval arithmetic.
% 
% An interval [2,3] is entered with the following notation.
>~2,3~
% Operators between intervals yield an interval result. The
% interval contains all possible results, when the operator
% is applied to arguments inside the intervals.
>~2,3~*~4,7~
% Note that the following are different. The first one
% contains all squares of elements in [-1,1], the second
% contains all s*t, where -1 <= s,t <=1.
>~-1,1~^2, ~-1,1~*~-1,1~,
% One may use the one point interval [s,s], but ~s~ is
% different from it. The latter is an interval around s.
>~2,2~, ~2~,
% Let us make an example. The 10-th partial sum of the
% Taylor series of exp is the following polynomial.
>p=1/fak(0:10)
% Let us evaluate this polynomial in -2.
>longformat; polyval(p,-2)
% The following is the correct result for the infinite series.
>exp(-2)
% We can get an inclusion with the remainder term in
% interval notation. Note that we have to use ~p~
% because p is not exactly represented in the computer.
% But we can use the one point interval [-2,-2].
>polyval(~p~,~-2,-2~)+~-1,1~*2^11/fak(11)
% Some more terms. Note that fak(21) is exactly representable
% in the computer.
>n=20; polyval(~1/fak(0:n)~,~-2,-2~)+~-1,1~*2^(n+1)/fak(n+1)
% You can see that interval arithmetic is a means of seeing
% how good a result really is.
% 
% Another example is the interval newton method. It is contained
% in the INTERVAL.E file. We need to load this file.
>load "interval"
% Furthermore, we need a function f and its derivative
% f1.c
>function f(x)
$return cos(x)-x
$endfunction
>function f1(x)
$return -sin(x)-1
$endfunction
% Then we can start the interval Newton method and
% get an inclusion of the result.
>inewton("f","f1",~0,1~)
% The usual Newton method delivers the same result.
>newton("f","f1",1)
% The Newton method accepts expressions in x for the
% function or for its derivative.
>newton("cos(x)-x","-sin(x)-1",1)
>inewton("cos(x)-x","-sin(x)-1",1)
% Let us try a multidimensional example.
>function f(x)
$return [x[1]^2+x[2]^2-1,x[1]^2/2+2*x[2]^2-1]
$endfunction
% To see how this function looks like, let us plot
% the contour lines of 0 of its components.
% 
% First we set up an array of x,y-values.
>x=-2:0.05:2; y=x';
% Then we plot the contour lines.
>contour(x^2+y^2-1,0); hold on; contour(x^2/2+2*y^2-1,0); hold off;
% Finally, we add a grid.
>setplot(-2,2,-2,2); xplot(); setplot();
% We need a derivative matrix.
>function df(x)
$return [2*x[1],2*x[2];x[1],4*x[2]]
$endfunction
% Now we are ready to get an inclusion of the
% intersection points (The zeros of f).
>inewton2("f","df",[~0.5,1~,~0.5,1~])
>newton2("f","df",[1,1])
>load "broyden"
% The Broyden method from UTIL.E does the same,
% but does not deliver a guaranteed inclusion.
% However, it does not need the derivative.
>broyden("f",[1,1])
% Let us make another example.
% 
% We search the interest rate of 26 payments.
>p=-1050|dup(100,12)'|dup(120,24)'|1000;
% We can plot the values with xmark.
>style("m*"); xmark(1:38,p);
% To solve the equation p(x), we define the function and
% the derivative.
>function g(x,p)
$return polyval(p,x);
$endfunction
% We pass p as a parameter to g and g1.
>function g1(x,p)
$return polyval(polydif(p),x);
$endfunction
% The polynomial p has a zero close to 0.9, as
% you can see.
>fplot("g",0,1;p);
% The Newton method yields a result, but we do
% not know how accurate it is.
>(1/newton("g","g1",1,p)-1)*100
% An interval method shows that the result is very accurate.
>(1/(inewton("g","g1",1,p))-1)*100
% We now solve a differential equation. The method
% is very elementary, but we get an inclusion of the
% solution.
% 
% The equation is y'=y/x^2. We enter it as a function.
>function D(x,y)
$return y/x^2
$endfunction
% We can now start the solver and get a very rough
% inclusion, the step size is 0.1.
>x=1:0.1:5; y=idgl("D",x,1); xplot(x,left(y)_right(y));
% With a step size of 0.01, the inclusion is better. For
% a change, we ener the equation as an expression.
>x=1:0.01:5; y=idgl("y/x^2",x,1); xplot(x,left(y)_right(y));
>
>