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diff --git a/src/input/images2a.input.pamphlet b/src/input/images2a.input.pamphlet new file mode 100644 index 00000000..05bd36c5 --- /dev/null +++ b/src/input/images2a.input.pamphlet @@ -0,0 +1,207 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input images2a.input} +\author{The Axiom Team} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{License} +<<license>>= +--Copyright The Numerical Algorithms Group Limited 1994. +@ +<<*>>= +<<license>> + +-- Newton's Iteration function + +-- newtonStep(f) returns a newton's iteration function for the +-- expression f. + +newtonStep(f) == + fun := complexNumericFunction f + deriv := complexDerivativeFunction(f,1) + (b:Complex DoubleFloat):Complex DoubleFloat +-> + b - fun(b)/deriv(b) + +-- create complex numeric functions from an expression + +complexFunPack := MakeUnaryCompiledFunction(EXPR INT, Complex DoubleFloat, Complex DoubleFloat) + +-- create a complex numeric function from an expression +complexNumericFunction x == + v := theVariable x + compiledFunction(x, v)$complexFunPack + +-- create a complex numeric derivatiave function from an expression +complexDerivativeFunction(x,n) == + v := theVariable x + df := differentiate(x,v,n) + compiledFunction(df, v)$complexFunPack + +-- return the unique variable in x, or an error if it is multivariate +theVariable x == + vl := variables x + nv := # vl + nv > 1 => error "Expression is not univariate." + nv = 0 => 'x + first vl + +-- complex surface and vector field drawing by SCM +-- complex surface vector field drawing + +C := Complex DoubleFloat +S := Segment DoubleFloat +PC := Record(rr:DoubleFloat, th:DoubleFloat) + +realSteps: PI := 25 -- the number of steps in the real direction +imagSteps: PI := 25 -- the number of steps in the imaginary direction +clipValue: DoubleFloat := 10 -- the maximum length of a vector to draw + + +-- Draw a complex function as a height field +-- uses the complex norm as the height and the complex argument as the color +-- optionally it will draw arrows on the surface indicating the direction +-- of the complex argument + +-- sample call: +-- f: C -> C +-- f z == exp(1/z) +-- drawComplex(f, 0.3..3, 0..2*%pi, false) + +-- parameter descriptions: +-- f: the function to draw +-- rRange: the range of the real values +-- imagRange: the range of imaginary values +drawComplex(f: C -> C, realRange: S, imagRange: S): VIEW3D == + free realSteps, imagSteps + delReal := (hi(realRange) - lo(realRange))/realSteps + delImag := (hi(imagRange) - lo(imagRange))/imagSteps + funTable: ARRAY2(PC) := new(realSteps+1, imagSteps+1, [0,0]$PC) + real := lo(realRange) + for i in 1..realSteps+1 repeat + imag := lo(imagRange) + for j in 1..imagSteps+1 repeat + z := f complex(real, imag) + funTable(i,j) := [clipFun(sqrt norm z), argument(z)]$PC + imag := imag + delImag + real := real + delReal + llp:List List Point DoubleFloat := [] + real := lo(realRange) + for i in 1..realSteps+1 repeat + imag := lo(imagRange) + lp:List Point DoubleFloat := [] + for j in 1..imagSteps+1 repeat + lp := cons(point [real,imag, funTable(i,j).rr, + funTable(i,j).th] ,lp) + imag := imag + delImag + real := real + delReal + llp := cons(reverse! lp, llp) + llp := reverse! llp + space := mesh(llp)$ThreeSpace(DoubleFloat) + makeViewport3D(space, "Complex Function")$VIEW3D + +-- draw a complex vector field +-- these vector fields should be viewed from the top by pressing the +-- "XY" translate button on the VIEW3D control panel + +-- parameters: +-- f: the mapping from C to C which we will draw +-- realRange: the range of the reals +-- tRange: the range of the imaginaries + +-- sample call: +-- f z == sin z +-- drawComplexVectorField(f, -2..2, -2..2) +-- call the functions 'setRealSteps' and 'setImagSteps' to change the +-- number of arrows drawn in each direction. + +drawComplexVectorField(f: C -> C, realRange: S, imagRange: S): VIEW3D == + -- compute the steps size of the grid + delReal := (hi(realRange) - lo(realRange))/realSteps + delImag := (hi(imagRange) - lo(imagRange))/imagSteps + -- create the space to hold the arrows + space := create3Space()$ThreeSpace DoubleFloat + real := lo(realRange) + for i in 1..realSteps+1 repeat + imag := lo(imagRange) + for j in 1..imagSteps+1 repeat + -- compute the function + z := f complex(real, imag) + -- get the direction of the arrow + arg := argument z + -- get the length of the arrow + len := clipFun(sqrt norm z) + -- create point at the base of the arrow + p1 := point [real, imag, 0.0@DoubleFloat, arg] + -- scale the arrow length so it isn't too long + scaleLen := delReal * len + -- create the point at the top of the arrow + p2 := point [p1.1 + scaleLen*cos(arg), p1.2 + scaleLen*sin(arg), + 0.0@DoubleFloat, arg] + -- make the pointer at the top of the arrow + arrow := makeArrow(p1, p2, scaleLen, arg) + -- add the line segments in the arrow to the space + for a in arrow repeat curve(space, a)$ThreeSpace DoubleFloat + imag := imag + delImag + real := real + delReal + -- draw the vector feild + makeViewport3D(space, "Complex Vector Field")$VIEW3D + +-- relative size of the arrow head compared to the length of the arrow +arrowScale := 0.25@DoubleFloat + +-- angle of the arrow head +arrowAngle := %pi-%pi/10.0@DoubleFloat + +-- Add an arrow head to a line segment, which starts at 'p1', ends at 'p2', +-- has length 'len', and and angle 'arg'. We pass 'len' and 'arg' as +-- arguments since thet were already computed by the calling program +makeArrow(p1, p2, len, arg) == + c1 := cos(arg + arrowAngle) + s1 := sin(arg + arrowAngle) + c2 := cos(arg - arrowAngle) + s2 := sin(arg - arrowAngle) + p3 := point [p2.1 + c1*arrowScale*len, p2.2 + s1*arrowScale*len, + p2.3, p2.4] + p4 := point [p2.1 + c2*arrowScale*len, p2.2 + s2*arrowScale*len, + p2.3, p2.4] + [[p1, p2, p3], [p2, p4]] + +-- set the number of steps to use in the real direction +setRealSteps(n) == + free realSteps + realSteps := n + +-- set the number of steps to use in the imaginary direction +setImagSteps(n) == + free imagSteps + imagSteps := n + +-- set the maximum length of a vector +setClipValue clip == + free clipValue + clipValue := clip + +-- clip a value in the interval (-clip...clip) +clipFun(x:DoubleFloat):DoubleFloat == + min(max(x, -clipValue), clipValue) + + +-- create a Newton's iteration function for the equation x**3 = 2. +f := newtonStep(x**3 - 2) + +setClipValue(4) +drawComplexVectorField(f**3, -3..3, -3..3) +drawComplex(f**3, -3..3, -3..3) +drawComplex(f**4, -3..3, -3..3) + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |