kill(all)$

maxima :polynoom operation,translation  version 1.0
peter.vlasschaert@gmail.com,19/11/2020

ref :youtube : https://www.youtube.com/watch?v=KhxiBgNfPBg
info :Bases of polynomial spaces | Wild Linear Algebra A 20 | NJ Wildberger

reason : relation between calculus & linear algebra.

load(lrats)$ ratprint:false$

powerdisp:true$

p1:p[α]:=sum(a[i]*α^i,i,0,3);

→ translation : polynoom 'thrid order'

p2:'p[x-3] = p[x-3];

p2a:lhs(p2)=expand(rhs(p2));

→ translation to right

p2b:lhs(p2)=sum(coeff(rhs(p2a),x,i)*x^i,i,0,3);

→ translation to left

p3:'p[x+3] = expand(p[x+3]);

p3a:lhs(p3)=sum(coeff(rhs(p3),x,i)*x^i,i,0,3);

p3b:x[i]:=coeff(rhs(p3a),x,i);

x[0];

p3ca:listofvars(%);

properties(p3ca);

p3ca1:apply(matrix,[p3ca]);

p3ca2:transpose(p3ca1);

p3cb:makelist(x[i],i,0,3);

p3d:coefmatrix(p3cb,p3ca);

p3dd:p3d[2,3];

simp:false;

p3da:p3d.p3ca2;

p3db:part(p3da,1);

simp:true;

p3dc:p3da;

p4:'p[α+c]=expand(p[α+c]);

p4a:lhs(p4)=sum(coeff(rhs(p4),α,i)*α^i,i,0,3);

y[i]:=coeff(rhs(p4a),α,i);

'y[2]=y[2];

depends(q,c);

p4b:q(c)=y[0];

p4b1:diff(p4b,c);

z[i]:=diff(p4b,c,i);

p5a:reverse(factorsum(z[0]));

p5a1:lhs(p5a)/y[0];

p5a2:p5a/p5a1;

p5b:reverse(factorsum(z[1]));

p5b1:lhs(p5b)/y[1];

p5b2:p5b/p5b1;

p5c:reverse(factorsum(z[2]));

p5c1:lhs(p5c)/y[2];

p5c2:p5c/p5c1;

p5d:reverse(factorsum(z[3]));

p5d1:lhs(p5d)/y[3];

p5d2:p5d/p5d1;

p6:[p5a2,p5b2,p5c2,p5d2];

p7:expand(lratsubst(p6,p4a));

rem : 0! =1, 1! =1,2! = 2,3!=6,4!=24

depends(qq,c);

p8:pp[c+α]=ev(rhs(p7),q=qq);

p9:p7*p8;

p10:expand(p9);

→ leibnitz : rules  'differentation'

p10a:lhs(p10)=sum(coeff(rhs(p10),α,i)*α^i,i,0,3);

→la = (l)inear (a)lgebra
==================

p10b:ratsimp(makelist(coeff(rhs(p10a),α,i),i,0,3));

→ca= calculus (differentation rules from calculus)
============

0 differentation,c

v1a:diff(p10b[1],c,0);/* divide by 0!=1,see above:la */

one times differentation,c

v1b:diff(p10b[1],c,1);/* divide by 1=1!,see above:la */

two times differentation,c

v1c:diff(p10b[1],c,2);/* divide by 2=2!,see above:la */

three times differentation,c

v1d:diff(p10b[1],c,3);/* divide by 6=3!,see above:la */

p11:'p[α+c]=p[α+c];

→  α :→ α-c

p11a:lratsubst([α = α-c],p11);

→ p7

p11b:p7;

→ taylor series

p11b1:ev(p11b,α=α-c);

→ c=0, Maclauren series

p11b2:at(p11b1,c=0);

basis
====

→ standard basis :[1,α,α^2,α^3]
→ factorial basis   :[1,α,α^2/2!,α^3/3!], for differentation  problems.