Árpád apánk

Blog. ennyi. semmi több. megtalalsz itt jó pár infót rólam, az életemről, meg néhány érdekes dolgot is, csak épp fejlesszem ki. addig is türelem kispajtás :)

2008/05/20

tizedik

> evalf(int(x^2+1,x=-1..1));

2.666666667

> f:=x->x^2+1;

f := proc (x) options operator, arrow; x^2+1 end proc

> ff:=evalf(f(-sqrt(3)/3)+f(sqrt(3)/3));

ff := 2.666666667

> evalf(int(x^3-1,x=0..2));

2.

> g:=x->x^3-1;

g := proc (x) options operator, arrow; x^3-1 end proc

> gg:=evalf(g((-sqrt(3)/3)+1)+g((sqrt(3)/3)+1));

gg := 1.999999999

> z:=x->cos(Pi*x);

z := proc (x) options operator, arrow; cos(Pi*x) end proc

> evalf(int(cos(Pi*x),x=0..1));

0.

> zz:=evalf((1/2)*((5/9)*z((-(-sqrt(15)/5)+1)/2))+(8/9)*z(1/2)+(5/9)*z((-(sqrt(15)/5)+1)/2));

zz := .2605477402

> n:=x->x^2+3*x+2;

n := proc (x) options operator, arrow; x^2+3*x+2 end proc

> diff(n(x),x);

2*x+3

> nn=expand((n(x+0.01)-n(x))/0.01);

nn = 3.01+2.*x

> m:=x->sin(x);

m := sin

> diff(diff(m(x),x),x);

-sin(x)

> mm:=expand((m(x+0.01)-2*m(x)+m(x-0.01))/0.01^2);

mm := -.99999*sin(x)

> xx:=[1,2,3,4,5];

xx := [1, 2, 3, 4, 5]

> yy:=[1,-3,-1,0,1];

yy := [1, -3, -1, 0, 1]

> c:=interp(xx,yy,x);

c := 1/3*x^4-9/2*x^3+65/3*x^2-85/2*x+26

> D(2.5)(c);

0

> px:=[Pi/2,Pi,3*Pi/2,2*Pi,5*Pi/2];

px := [Pi/2, Pi, 3*Pi/2, 2*Pi, 5*Pi/2]

> py:=[1,0,-1,0,1];

py := [1, 0, -1, 0, 1]

> d:=interp(px,py,x);

d := -8*x^4/(3*Pi^4)+16*x^3/Pi^3-94*x^2/(3*Pi^2)+22*x/Pi-4

> D(Pi)(d);

0

> with(CurveFitting);

>

[BSpline, BSplineCurve, Interactive, LeastSquares, PolynomialInterpolation, RationalInterpolation, Spline, ThieleInterpolation]

> PolynomialInterpolation(px,py,x,Newton);

-8*x^4/(3*Pi^4)+16*x^3/Pi^3-94*x^2/(3*Pi^2)+22*x/Pi-4

> with(plots);

Warning, the name changecoords has been redefined

[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...

> h:=[[x[0],y(x[0])]];
for i from 0 to 9 do
x[0]:=0;
y(x[0]):=2;

x[i+1]:=x[i]+0.1;
y(x[i+1]):=y(x[i])+0.1*(1+x[i]^2);
h:=[op(h),[x[i+1],y(x[i+1])]];
end do;

h := [[0, 2]]

x[0] := 0

y(0) := 2

x[1] := .1

y(.1) := 2.1

h := [[0, 2], [.1, 2.1]]

x[0] := 0

y(0) := 2

x[2] := .2

y(.2) := 2.201

h := [[0, 2], [.1, 2.1], [.2, 2.201]]

x[0] := 0

y(0) := 2

x[3] := .3

y(.3) := 2.305

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305]]

x[0] := 0

y(0) := 2

x[4] := .4

y(.4) := 2.414

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414]]

x[0] := 0

y(0) := 2

x[5] := .5

y(.5) := 2.530

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414], [.5, 2.530]]

x[0] := 0

y(0) := 2

x[6] := .6

y(.6) := 2.655

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414], [.5, 2.530], [.6, 2.655]]

x[0] := 0

y(0) := 2

x[7] := .7

y(.7) := 2.791

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414], [.5, 2.530], [.6, 2.655], [.7, 2.791]]

x[0] := 0

y(0) := 2

x[8] := .8

y(.8) := 2.940

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414], [.5, 2.530], [.6, 2.655], [.7, 2.791], [.8, 2.940]]

x[0] := 0

y(0) := 2

x[9] := .9

y(.9) := 3.104

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414], [.5, 2.530], [.6, 2.655], [.7, 2.791], [.8, 2.940], [.9, 3.104]]

x[0] := 0

y(0) := 2

x[10] := 1.0

y(1.0) := 3.285

h := [[0, 2], [.1, 2.1], [.2, 2.201], [.3, 2.305], [.4, 2.414], [.5, 2.530], [.6, 2.655], [.7, 2.791], [.8, 2.940], [.9, 3.104], [1.0, 3.285]]

> pointplot(h);

[Plot]

>

posted by Miklós Árpád @ 07:58  0 comments

kilencedik

> with(student);

[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, completesquare, distance, equate, integrand, intercept, intparts, leftbox, leftsum, makeproc, middlebox, middlesum, midpoi...[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, completesquare, distance, equate, integrand, intercept, intparts, leftbox, leftsum, makeproc, middlebox, middlesum, midpoi...

> f:=x->x^7*ln(x)+x-cos(x);

f := proc (x) options operator, arrow; x^7*ln(x)+x-cos(x) end proc

> g:=x->exp(-x^2);

g := proc (x) options operator, arrow; exp(-x^2) end proc

> h:=x->exp(x)*sin(x);

h := proc (x) options operator, arrow; exp(x)*sin(x) end proc

> evalf(int(f(x),x=2..3));

783.5712461

> evalf(int(g(x),x=-1..1));

1.493648266

> evalf(int(h(x),x=0..Pi));

12.07034632

> middlebox(f(x),x=2..3);

[Plot]

> evalf(middlesum(f(x),x=2..3,10));

781.0907780

> evalf(simpson(f(x),x=2..3,100));

783.5712470

> middlebox(g(x),x=-1..1,10);

[Plot]

> evalf(middlesum(g(x),x=-1..1));

1.509195888

> evalf(simpson(g(x),x=-1..1,100));

1.493648269

> middlebox(h(x),x=0..Pi,10);

[Plot]

> evalf(middlesum(h(x),x=0..Pi));

12.66778405

> evalf(simpson(h(x),x=0..Pi,100));

12.07034606

> teglalap:=proc(f,a,b,n)
local h,d;
h:=(b-a)/n;
d:=(b-a)/n*sum(f(a-h/2+i*h),i=1..n);
return evalf(d);
end proc;

>

teglalap := proc (f, a, b, n) local h, d; h := (b-a)/n; d := (b-a)*(sum(f(a-1/2*h+i*h), i = 1 .. n))/n; return evalf(d) end proc

> teglalap(f,2,3,10);

781.0907781

> with(Student[Calculus1]);

[AntiderivativePlot, ApproximateInt, ArcLength, Asymptotes, Clear, CriticalPoints, DerivativePlot, ExtremePoints, FunctionAverage, FunctionChart, GetMessage, GetNumProblems, GetProblem, Hint, Inflecti...[AntiderivativePlot, ApproximateInt, ArcLength, Asymptotes, Clear, CriticalPoints, DerivativePlot, ExtremePoints, FunctionAverage, FunctionChart, GetMessage, GetNumProblems, GetProblem, Hint, Inflecti...[AntiderivativePlot, ApproximateInt, ArcLength, Asymptotes, Clear, CriticalPoints, DerivativePlot, ExtremePoints, FunctionAverage, FunctionChart, GetMessage, GetNumProblems, GetProblem, Hint, Inflecti...

> ApproximateInt(f(x),x=2..3);

5/2+271818611107/12800000000*ln(43/20)-1/10*cos(9/4)-1/10*cos(43/20)-1/10*cos(41/20)-1/10*cos(11/4)-1/10*cos(53/20)-1/10*cos(51/20)-1/10*cos(49/20)-1/10*cos(47/20)-1/10*cos(59/20)-1/10*cos(57/20)+2488...5/2+271818611107/12800000000*ln(43/20)-1/10*cos(9/4)-1/10*cos(43/20)-1/10*cos(41/20)-1/10*cos(11/4)-1/10*cos(53/20)-1/10*cos(51/20)-1/10*cos(49/20)-1/10*cos(47/20)-1/10*cos(59/20)-1/10*cos(57/20)+2488...

>

posted by Miklós Árpád @ 07:58  0 comments

nyolcadik

> with(CurveFitting);

>

[BSpline, BSplineCurve, Interactive, LeastSquares, PolynomialInterpolation, RationalInterpolation, Spline, ThieleInterpolation]

> expand(PolynomialInterpolation([0,1,2,3],[1,3,-1,2],x,form=Lagrange));

>

>

13/6*x^3-19/2*x^2+28/3*x+1

> lag:=expand(PolynomialInterpolation([0,1,2,3],[1,3,-1,2],x,form=Newton));

>

lag := 13/6*x^3-19/2*x^2+28/3*x+1

> pointplot({[0,1],[1,3],[2,-1],[3,2]},symbolsize=25);

[Plot]

> display(pointplot({[0,1],[1,3],[2,-1],[3,2]},symbolsize=25),plot(lag,x=-1..4,y=-5..5));

[Plot]

> with(plots);

Warning, the name changecoords has been redefined

[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...

> xi:=[0,1,3,4.2,5,5.8,7,8];

>

xi := [0, 1, 3, 4.2, 5, 5.8, 7, 8]

> yi:=[1,2.5,2.3,5,3.5,4.2,7,10];

yi := [1, 2.5, 2.3, 5, 3.5, 4.2, 7, 10]

> sp1:=Spline(xi,yi,x);

sp1 := PIECEWISE([1.+2.04045418599999984*x-.540454185700000034*x^3, x < 1], [2.080908371+.419091629299999990*x-1.62136255679615826*(x-1)^2+.680908371300000037*(x-1)^3, x < 3], [-4.013625568+2.10454185...

> display(pointplot({[0,1],[1,2.5],[3,2.3],[4.2,5],[5,3.5],[5.8,4.2],[7,7],[8,10]},symbolsize=25),plot(sp1,x,y=-1..12,color=green),plot(lag1,y=-5..12,x=-4..4,color=blue));

[Plot]

> lag1:=expand(PolynomialInterpolation(xi,yi,x,form=Lagrange));

lag1 := 47.66577892*x+59.68279433*x^3-89.75801475*x^2-.2626950677*x^6-18.98393180*x^4+3.147340024*x^5+0.8728350885e-2*x^7+1.000000000

> f:=x->(cos(Pi*x))^3-4*x;

f := proc (x) options operator, arrow; cos(Pi*x)^3-4*x end proc

> xx:=[-1,-1/2,0,1/2,1];

xx := [-1, (-1)/2, 0, 1/2, 1]

> g:=x->A*x^3+B*x^2+C*x+D;

g := proc (x) options operator, arrow; A*x^3+B*x^2+C*x+D end proc

> y1:=map(f,xx);

y1 := [3, 2, 1, -2, -5]

> osszeg:=(A,B,C,D)->sum((y1[i]-(g(xx[i])))^2,i=1..5);

osszeg := proc (A, B, C, D) options operator, arrow; sum((y1[i]-g(xx[i]))^2, i = 1 .. 5) end proc

> DA:=diff(osszeg(A,B,C,D),A);

DA := 17+65*A/16+17*C/4

> DB:=diff(osszeg(A,B,C,D),B);

DB := 4+17*B/4+5*D

> DC:=diff(osszeg(A,B,C,D),C);

DC := 20+17*A/4+5*C

> DD:=diff(osszeg(A,B,C,D),D);

DD := 2+5*B+10*D

> eqA:=DA=0;

>

eqA := 17+65*A/16+17*C/4 = 0

> eqB:=DB=0;

eqB := 4+17*B/4+5*D = 0

> eqC:=DC=0;

eqC := 20+17*A/4+5*C = 0

> eqD:=DD=0;

eqD := 2+5*B+10*D = 0

>

> megold:=solve({eqA,eqB,eqC,eqD},{A,B,C,D});

megold := {C = -4, A = 0, D = 23/35, B = (-12)/7}

> polinom:=subs(megold,g(x));

polinom := 23/35-12/7*x^2-4*x

> LeastSquares(xx,y1,x,curve=A*x^3+B*x^2+C*x+D);

23/35-12/7*x^2-4*x

> with(stats);

[anova, describe, fit, importdata, random, statevalf, statplots, transform]

> with(statplots);

[boxplot, histogram, scatterplot, xscale, xshift, xyexchange, xzexchange, yscale, yshift, yzexchange, zscale, zshift]

> scatterplot(xx,y1,symbolsize=25);

[Plot]

> c:=scatterplot(xx,y1,symbolsize=25):

> display(c,plot(polinom,x=-1..1));

[Plot]

>

posted by Miklós Árpád @ 07:56  0 comments

hetedik

> herm:=x->a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5;

herm := proc (x) options operator, arrow; a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5 end proc

> x0:=-1;

x0 := -1

> x1:=0;

x1 := 0

> x2:=1;

x2 := 1

> herm(x0)=1;

a0-a1+a2-a3+a4-a5 = 1

> herm(x1)=1;

a0 = 1

> herm(x2)=-2;

a0+a1+a2+a3+a4+a5 = -2

> hermder:=x->a1+2*a2*x+3*a3*x^2+4*a4*x^3+5*a5*x^4;

>

hermder := proc (x) options operator, arrow; a1+2*a2*x+3*a3*x^2+4*a4*x^3+5*a5*x^4 end proc

> hermder(x0)=2;

a1-2*a2+3*a3-4*a4+5*a5 = 2

> hermder(x1)=3;

a1 = 3

> hermder(x2)=0;

a1+2*a2+3*a3+4*a4+5*a5 = 0

> b:=solve({herm(x0)=1,herm(x1)=1,herm(x2)=-2,hermder(x0)=2,hermder(x1)=3,hermder(x2)=0});

b := {a0 = 1, a1 = 3, a5 = 23/4, a4 = 1, a3 = (-41)/4, a2 = (-5)/2}

> c:=subs(b,herm(x));

c := 1+3*x-5/2*x^2-41/4*x^3+x^4+23/4*x^5

> plot(c,x=-1..1);

[Plot]

>

> newtonpolegyutt:=proc(xk,yk)
local n,i,j;
n:=nops(xk)-1;
for i from 0 to n do
a[i+1]:=yk[i+1];
end do;
for i from 1 to n do
for j from n to i by -1 do
a[j+1]:=(a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]);
end do;
end do;
return op(a);
end proc;

>

Warning, `a` is implicitly declared local to procedure `newtonpolegyutt`

newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...newtonpolegyutt := proc (xk, yk) local n, i, j, a; n := nops(xk)-1; for i from 0 to n do a[i+1] := yk[i+1] end do; for i to n do for j from n by -1 to i do a[j+1] := (a[j+1]-a[j])/(xk[j+1]-xk[j-i+1]) ...

> xk:=[0.0,0.1,0.2,0.3,0.4,0.5];

xk := [0., .1, .2, .3, .4, .5]

> yk:=map(sin,xk);

yk := [0., 0.9983341665e-1, .1986693308, .2955202067, .3894183423, .4794255386]

> a:=newtonpolegyutt(xk,yk);

a := TABLE([1 = 0., 2 = .9983341665, 3 = -0.4987512500e-1, 4 = -.1645892917, 5 = 0.8264041750e-2, 6 = 0.8057791500e-2])

> newtonpol:=proc(xk,a,x)
local p,i,n;
n:=nops(xk)-1;
p:=a[n+1];
for i from n-1 to 0 by -1 do
p:=a[i+1]+(x-xk[i+1])*p;
end do;
return p;
end proc;

newtonpol := proc (xk, a, x) local p, i, n; n := nops(xk)-1; p := a[n+1]; for i from n-1 by -1 to 0 do p := a[i+1]+(x-xk[i+1])*p end do; return p end proc

> newtonpol(xk,a,0.45);

.4349655284

> sin(0.45);

.4349655341

> f:=x->exp(2*x);

f := proc (x) options operator, arrow; exp(2*x) end proc

> xk:=[seq(-1+k/4,k=0..8)];

xk := [-1, (-3)/4, (-1)/2, (-1)/4, 0, 1/4, 1/2, 3/4, 1]

> yk:=map(f,xk);

yk := [exp(-2), exp((-3)/2), exp(-1), exp((-1)/2), 1, exp(1/2), exp(1), exp(3/2), exp(2)]

> a:=evalf(newtonpolegyutt(xk,yk));

a := TABLE([1 = .1353352832, 2 = .3511795076, 3 = .455635234, 4 = .394107020, 5 = .255665620, 6 = .132684572, 7 = 0.573835179e-1, 8 = 0.212719256e-1, 9 = 0.68997796e-2])

> expand(newtonpol(xk,a,x));

1.000000001+1.999986730*x+1.999997393*x^2+1.333635260*x^3+.6667263733*x^4+.2650666476*x^5+0.8857208056e-1*x^6+0.2817170520e-1*x^7+0.68997796e-2*x^8

>

posted by Miklós Árpád @ 07:55  0 comments

hatodik

> f:=x->x*cos(x);

f := proc (x) options operator, arrow; x*cos(x) end proc

> z:=interp([0,2*Pi/3,4*Pi/3,2*Pi],[f(0),f(2*Pi/3),f(4*Pi/3),f(2*Pi)],x);

z := 27*x^3/(16*Pi^2)-27*x^2/(8*Pi)+x

> evalf(z);

.1709794973*x^3-1.074295866*x^2+x

> plot(f);

[Plot]

> with(plots);

Warning, the name changecoords has been redefined

[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...

> gr1:=plot(f(x),x=0..2*Pi,color=red):

> gr2:=plot(z,x=0..2*Pi,color=green):

> display(gr1,gr2);

[Plot]

> lagrange:=proc(x,y,a)
local n,v,i,j,t;
v:=0;
n:=nops(x);
for i from 0 to n-1 do
t:=y[i+1];
for j from 0 by 1 to i-1 do
t:=t*(a-x[j+1])/(x[i+1]-x[j+1]);
end do;
for j from i+1 to n-1 do
t:=t*(a-x[j+1])/(x[i+1]-x[j+1]);
end do;
v:=v+t;
end do;
end proc;

>

lagrange := proc (x, y, a) local n, v, i, j, t; v := 0; n := nops(x); for i from 0 to n-1 do t := y[i+1]; for j from 0 to i-1 do t := t*(a-x[j+1])/(x[i+1]-x[j+1]) end do; for j from i+1 to n-1 do t :=...lagrange := proc (x, y, a) local n, v, i, j, t; v := 0; n := nops(x); for i from 0 to n-1 do t := y[i+1]; for j from 0 to i-1 do t := t*(a-x[j+1])/(x[i+1]-x[j+1]) end do; for j from i+1 to n-1 do t :=...lagrange := proc (x, y, a) local n, v, i, j, t; v := 0; n := nops(x); for i from 0 to n-1 do t := y[i+1]; for j from 0 to i-1 do t := t*(a-x[j+1])/(x[i+1]-x[j+1]) end do; for j from i+1 to n-1 do t :=...lagrange := proc (x, y, a) local n, v, i, j, t; v := 0; n := nops(x); for i from 0 to n-1 do t := y[i+1]; for j from 0 to i-1 do t := t*(a-x[j+1])/(x[i+1]-x[j+1]) end do; for j from i+1 to n-1 do t :=...lagrange := proc (x, y, a) local n, v, i, j, t; v := 0; n := nops(x); for i from 0 to n-1 do t := y[i+1]; for j from 0 to i-1 do t := t*(a-x[j+1])/(x[i+1]-x[j+1]) end do; for j from i+1 to n-1 do t :=...lagrange := proc (x, y, a) local n, v, i, j, t; v := 0; n := nops(x); for i from 0 to n-1 do t := y[i+1]; for j from 0 to i-1 do t := t*(a-x[j+1])/(x[i+1]-x[j+1]) end do; for j from i+1 to n-1 do t :=...

> expand(lagrange([0,2*Pi/3,4*Pi/3,2*Pi],[f(0),f(2*Pi/3),f(4*Pi/3),f(2*Pi)],x));

27*x^3/(16*Pi^2)-27*x^2/(8*Pi)+x

> t:=(b+a)/2-(b-a)/2*cos(2*k-1)*Pi/(2*n);

t := b/2+a/2-1/4*(b-a)*cos(2*k-1)*Pi/n

> f2:=x->sin(x)+cos(x);

f2 := proc (x) options operator, arrow; sin(x)+cos(x) end proc

> t1:=(2*Pi+0)/2-(2*Pi-0)/2*cos((2*1-1)*Pi/(2*4));

t1 := Pi-Pi*cos(Pi/8)

> t2:=(2*Pi+0)/2-(2*Pi-0)/2*cos((2*2-1)*Pi/(2*4));

t2 := Pi-Pi*cos(3*Pi/8)

> t3:=(2*Pi+0)/2-(2*Pi-0)/2*cos((2*3-1)*Pi/(2*4));

t3 := Pi+Pi*cos(3*Pi/8)

> t4:=(2*Pi+0)/2-(2*Pi-0)/2*cos((2*4-1)*Pi/(2*4));

t4 := Pi+Pi*cos(Pi/8)

> c:=evalf(interp([t1,t2,t3,t4],[f(t1),f(t2),f(t3),f(t4)],x));

c := 1.247370174*x+.1908355625*x^3-1.199055202*x^2+0.16e-7

> gr1:=plot(f(x),x=0..2*Pi,color=red):

> gr2:=plot(z,x=0..2*Pi,color=green):

> gr3:=plot(c,x=0..2*Pi,color=blue):

> display(gr1,gr2,gr3);

[Plot]

> with(numapprox);

[chebdeg, chebmult, chebpade, chebsort, chebyshev, confracform, hermite_pade, hornerform, infnorm, laurent, minimax, pade, remez]

> plot(ChebyshevT(5,x),x=-1..1);

[Plot]

>

posted by Miklós Árpád @ 07:54  0 comments