十進BASIC第2掲示板過去ログ

X^2+Y^2-3XY+1=0のグラフ

投稿者：しばっち投稿日：2020年 4月 5日(日)14時27分39秒

X^2+Y^2-3*X*Y+1=0のグラフ https://hp.vector.co.jp/authors/VA008683/F_GRAPH.htm 導関数 dy/dx=(2*x-3*y)/(3*x-2*y)を求め、dx/dt=3*x-2*y dy/dt=2*x-3*yとして 2元連立常微分方程式をルンゲクッタ法で解く https://www.wolframalpha.com/input/?i=x%5E2%2By%5E2-3*x*y%2B1%3D0%2Cdy%2Fdx&lang=ja 陰関数 F(x,y)=0のグラフはdy/dx=f(x,y)の解曲線とすれば描画できる。 LET XS=-5 LET XE=5 LET YS=-5 LET YE=5 SET WINDOW XS,XE,YS,YE DRAW GRID LET N=100 !'分割数 LET H=(XE-XS)/N FOR L=1 TO 2 READ XX,YY DATA 1,1 DATA -1,-1 FOR I=1 TO 2 LET T=0 LET X=XX !'初期値 LET Y=YY WHEN EXCEPTION IN FOR J=1 TO N LET K1=F1(T,X,Y) LET L1=F2(T,X,Y) LET K2=F1(T+H/2,X+H/2*K1,Y+H/2*L1) LET L2=F2(T+H/2,X+H/2*K1,Y+H/2*L1) LET K3=F1(T+H/2,X+H/2*K2,Y+H/2*L2) LET L3=F2(T+H/2,X+H/2*K2,Y+H/2*L2) LET K4=F1(T+H,X+H*K3,Y+H*L3) LET L4=F2(T+H,X+H*K3,Y+H*L3) LET X0=X+H*(K1+2*K2+2*K3+K4)/6 LET Y0=Y+H*(L1+2*L2+2*L3+L4)/6 LET T0=T+H PLOT LINES:X,Y;X0,Y0 LET X=X0 LET Y=Y0 LET T=T0 NEXT J USE PLOT LINES END WHEN LET H=-H NEXT I NEXT L END EXTERNAL FUNCTION F1(T,X,Y) !'dx/dt=f1(t,x,y) LET F1=3*X-2*Y END FUNCTION EXTERNAL FUNCTION F2(T,X,Y) !'dy/dt=f2(t,x,y) LET F2=2*X-3*Y END FUNCTION -------------------------------------------------------------------------------- 1/(X-Y)-X+3*Y=0のグラフ dy/dx=((x-y)^2+1)/(3*(x-y)^2+1) https://www.wolframalpha.com/input/?i=1%2F%28x-y%29-x%2B3*y%3D0%2Cdy%2Fdx&lang=ja LET XS=-5 LET XE=5 LET YS=-5 LET YE=5 SET WINDOW XS,XE,YS,YE DRAW GRID LET N=100 !'分割数 LET H=(XE-XS)/N FOR L=1 TO 2 READ XX,YY DATA 1,0 DATA -1,0 FOR I=1 TO 2 LET T=0 LET X=XX !'初期値 LET Y=YY WHEN EXCEPTION IN FOR J=1 TO N LET K1=F1(T,X,Y) LET L1=F2(T,X,Y) LET K2=F1(T+H/2,X+H/2*K1,Y+H/2*L1) LET L2=F2(T+H/2,X+H/2*K1,Y+H/2*L1) LET K3=F1(T+H/2,X+H/2*K2,Y+H/2*L2) LET L3=F2(T+H/2,X+H/2*K2,Y+H/2*L2) LET K4=F1(T+H,X+H*K3,Y+H*L3) LET L4=F2(T+H,X+H*K3,Y+H*L3) LET X0=X+H*(K1+2*K2+2*K3+K4)/6 LET Y0=Y+H*(L1+2*L2+2*L3+L4)/6 LET T0=T+H PLOT LINES:X,Y;X0,Y0 LET X=X0 LET Y=Y0 LET T=T0 NEXT J USE PLOT LINES END WHEN LET H=-H NEXT I NEXT L END EXTERNAL FUNCTION F1(T,X,Y) !'dx/dt=f1(t,x,y) LET F1=3*(X-Y)^2+1 END FUNCTION EXTERNAL FUNCTION F2(T,X,Y) !'dy/dt=f2(t,x,y) LET F2=(X-Y)^2+1 END FUNCTION

Re: X^2+Y^2-3XY+1=0のグラフ

投稿者：しばっち投稿日：2020年 4月 8日(水)20時36分41秒

> No.4817[元記事へ] 陰関数 F(x,y)=0 の導関数 dy/dx=f(x,y) は数値微分でも代用できるようです。 LET XS=-5 LET XE=5 LET YS=-5 LET YE=5 SET WINDOW XS,XE,YS,YE DRAW GRID LET N=100 !'分割数 LET H=(XE-XS)/N FOR L=1 TO 2 READ XX,YY DATA 1,1 DATA -1,-1 FOR I=1 TO 2 LET T=0 LET X=XX !'初期値 LET Y=YY WHEN EXCEPTION IN FOR J=1 TO N LET K1=F1(T,X,Y) LET L1=F2(T,X,Y) LET K2=F1(T+H/2,X+H/2*K1,Y+H/2*L1) LET L2=F2(T+H/2,X+H/2*K1,Y+H/2*L1) LET K3=F1(T+H/2,X+H/2*K2,Y+H/2*L2) LET L3=F2(T+H/2,X+H/2*K2,Y+H/2*L2) LET K4=F1(T+H,X+H*K3,Y+H*L3) LET L4=F2(T+H,X+H*K3,Y+H*L3) LET X0=X+H*(K1+2*K2+2*K3+K4)/6 LET Y0=Y+H*(L1+2*L2+2*L3+L4)/6 LET T0=T+H PLOT LINES:X,Y;X0,Y0 LET X=X0 LET Y=Y0 LET T=T0 IF X>XE OR X<XS OR Y<YS OR Y>YE THEN EXIT FOR NEXT J USE PLOT LINES END WHEN LET H=-H NEXT I NEXT L END EXTERNAL FUNCTION F1(T,X,Y) !'dx/dt=f1(t,x,y) LET F1=FY(X,Y) END FUNCTION EXTERNAL FUNCTION F2(T,X,Y) !'dy/dt=f2(t,x,y) LET F2=-FX(X,Y) END FUNCTION EXTERNAL FUNCTION F(X,Y) !'陰関数 F(x,y)=0 LET F=X^2+Y^2-3*X*Y+1 END FUNCTION EXTERNAL FUNCTION FX(X,Y) !' ∂/∂x F(x,y) LET H=1/256 LET FX=(F(X-2*H,Y)-8*F(X-H,Y)+8*F(X+H,Y)-F(X+2*H,Y))/(12*H) END FUNCTION EXTERNAL FUNCTION FY(X,Y) !' ∂/∂y F(x,y) LET H=1/256 LET FY=(F(X,Y-2*H)-8*F(X,Y-H)+8*F(X,Y+H)-F(X,Y+2*H))/(12*H) END FUNCTION

Re: X^2+Y^2-3XY+1=0のグラフ

投稿者：しばっち投稿日：2020年 4月12日(日)14時46分5秒

> No.4818[元記事へ] 下記のようにしても陰関数のグラフが描画できます。 LET LEFT=-5 LET RIGHT=5 LET BOTTOM=-5 LET TOP=5 SET POINT STYLE 1 SET WINDOW LEFT,RIGHT,BOTTOM,TOP ASK PIXEL SIZE XSIZE,YSIZE DRAW GRID FOR YY=0 TO YSIZE-1 FOR XX=0 TO YSIZE-1 LET X=WORLDX(XX) LET Y=WORLDY(YY) WHEN EXCEPTION IN LET L=ABS(F(X,Y)) IF L<1 AND L/SQR(FX(X,Y)^2+FY(X,Y)^2)<.01 THEN PLOT POINTS: X,Y USE END WHEN NEXT XX NEXT YY END EXTERNAL FUNCTION F(X,Y) !'陰関数 F(X,Y)=0 LET F=X*X+Y*Y-3*X*Y+1 !'LET F=1/(X-Y)-X+3*Y !'LET F=SIN(X)+SIN(2*Y)+SIN(3*X)+SIN(4*Y)+SIN(5*X) !'LET F=(FF(Y)^2-FF(X)^4+FF(X)^6)*(FF(X)^2-FF(Y)^4+FF(Y)^6) !'LET F=(FF(X)^2+FF(Y)^2-3)*(1.5*FF(X)^4-FF(X)^6-1.5*FF(Y)^4) !'LET F=(FF(X)^2+FF(Y)^2)^3-4*FF(X)^2*FF(Y)^2 END FUNCTION EXTERNAL FUNCTION FF(X) LET FF=X-2.6*INT((X+1.3)/2.6) END FUNCTION EXTERNAL FUNCTION FX(X,Y) !' ∂/∂x F(x,y) LET H=1/256 !'LET FX=(-F(X-H,Y)+F(X+H,Y))/(2*H) LET FX=(F(X-2*H,Y)-8*F(X-H,Y)+8*F(X+H,Y)-F(X+2*H,Y))/(12*H) !'LET FX=(-F(X-3*H,Y)+9*F(X-2*H,Y)-45*F(X-H,Y)+45*F(X+H,Y)-9*F(X+2*H,Y)+F(X+3*H,Y))/(60*H) !'LET FX=(3*F(X-4*H,Y)-32*F(X-3*H,Y)+168*F(X-2*H,Y)-672*F(X-H,Y)+672*F(X+H,Y)-168*F(X+2*H,Y)+32*F(X+3*H,Y)-3*F(X+4*H,Y))/(840*H) END FUNCTION EXTERNAL FUNCTION FY(X,Y) !' ∂/∂y F(x,y) LET H=1/256 !'LET FY=(-F(X,Y-H)+F(X,Y+H))/(2*H) LET FY=(F(X,Y-2*H)-8*F(X,Y-H)+8*F(X,Y+H)-F(X,Y+2*H))/(12*H) !'LET FY=(-F(X,Y-3*H)+9*F(X,Y-2*H)-45*F(X,Y-H)+45*F(X,Y+H)-9*F(X,Y+2*H)+F(X,Y+3*H))/(60*H) !'LET FY=(3*F(X,Y-4*H)-32*F(X,Y-3*H)+168*F(X,Y-2*H)-672*F(X,Y-H)+672*F(X,Y+H)-168*F(X,Y+2*H)+32*F(X,Y+3*H)-3*F(X,Y+4*H))/(840*H) END FUNCTION

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十進BASIC 第2掲示板 過去ログ

X^2+Y^2-3*X*Y+1=0のグラフ

Re: X^2+Y^2-3*X*Y+1=0のグラフ

Re: X^2+Y^2-3*X*Y+1=0のグラフ

十進BASIC 第2掲示板過去ログ

X^2+Y^2-3XY+1=0のグラフ

Re: X^2+Y^2-3XY+1=0のグラフ

Re: X^2+Y^2-3XY+1=0のグラフ