jknuba_NM

D:\jknuba_2025_spring\jknuba_NM\jknuba_NM_Job6.wxmx

Метод ітерацій (Method of iterations)

(%i1)

kill(all);

\[\]\[\tag{%o0} \ensuremath{\mathrm{done}}\]

(%i1)

M(x):=0.9·x;

\[\]\[\tag{%o1} \mathop{M}(x)\mathop{:=}0.9 x\]

(%i2)

N(x):=2.2·(x+1)−5;

\[\]\[\tag{%o2} \mathop{N}(x)\mathop{:=}2.2 \left( x\mathop{+}1\right) \mathop{-}5\]

(%i3)

X(n):=((n+5)/2.2)−1;

\[\]\[\tag{%o3} \mathop{X}(n)\mathop{:=}\frac{n\mathop{+}5}{2.2}\mathop{-}1\]

(%i4)

R(x):=N(x)−M(x);

\[\]\[\tag{%o4} \mathop{R}(x)\mathop{:=}\mathop{N}(x)\mathop{-}\mathop{M}(x)\]

(%i32)

plot2d([M(x),N(x)],[x,0,5],
[legend, "M(x)","N(x)"],
[style, [lines, 3,5], [lines, 3,1]]);

\[\]\[\tag{%o32} false\]

Figure 1:

Iteration 1

(%i6)

delta:0.1;

\[\]\[\tag{%o6} 0.1\]

(%i7)

i:0;

\[\]\[\tag{%o7} 0\]

(%i9)

m:M(i);o:X(m);

\[\]\[\tag{%o8} 0\]

\[\]\[\tag{%o9} 1.2727272727272725\]

(%i10)

if (M(o)−N(o)<delta)then display("OK")else display(M(o)−N(o));

\[\]\[1.1454545454545453\mathop{+}-0.0\mathop{=}1.1454545454545453\]

\[\]\[\tag{%o10} \ensuremath{\mathrm{done}}\]

(%i12)

o;N(o);

\[\]\[\tag{%o11} 1.2727272727272725\]

\[\]\[\tag{%o12} 0.0\]

(%i13)

plot2d([M(x),N(x),[discrete,[i,o],[M(i),N(o)]], [discrete,[o],[N(o)]]],[x,0,5],
[legend, "M(x)","N(x)", "Iteration", "point"],
[style, [lines, 3,5], [lines, 3,1], [lines, 2,2], [points,6,1 ]]);

\[\]\[\tag{%o13} false\]

Figure 2:

Iteration 2

(%i14)

plot2d([M(x),N(x),[discrete,[i,o],[M(i),N(o)]],[discrete,[o,o],[M(o),N(o)]],[discrete,[o],[N(o)]]],[x,0,5],
[legend, "M(x)","N(x)", "Iteration","Iteration_2", "point"],
[style, [lines, 3,5], [lines, 3,1], [lines, 2,2],[lines, 5,3], [points,6,1 ]])$

Figure 3:

(%i15)

i:o;

\[\]\[\tag{%o15} 1.2727272727272725\]

(%i17)

m:M(i);o:X(m);

\[\]\[\tag{%o16} 1.1454545454545453\]

\[\]\[\tag{%o17} 1.7933884297520657\]

(%i18)

if (M(o)−N(o)<delta)then display("OK")else display(M(o)−N(o));

\[\]\[1.6140495867768592\mathop{+}\mathop{-}1.1454545454545446\mathop{=}0.46859504132231455\]

\[\]\[\tag{%o18} \ensuremath{\mathrm{done}}\]

(%i19)

plot2d([M(x),N(x),[discrete,[i,o],[M(i),N(o)]],[discrete,[o,o],[M(o),N(o)]],[discrete,[o],[N(o)]]],[x,0,5],
[legend, "M(x)","N(x)", "Iteration_2","Iteration_3", "point"],
[style, [lines, 3,5], [lines, 3,1], [lines, 2,2],[lines, 5,3], [points,6,1 ]])$

Figure 4:

Iteration 3

(%i20)

i:o;

\[\]\[\tag{%o20} 1.7933884297520657\]

(%i22)

m:M(i);o:X(m);

\[\]\[\tag{%o21} 1.6140495867768592\]

\[\]\[\tag{%o22} 2.006386175807663\]

(%i23)

if (M(o)−N(o)<delta)then display("OK")else display(M(o)−N(o));

\[\]\[1.805747558226897\mathop{+}\mathop{-}1.6140495867768596\mathop{=}0.1916979714500373\]

\[\]\[\tag{%o23} \ensuremath{\mathrm{done}}\]

(%i24)

plot2d([M(x),N(x),[discrete,[i,o],[M(i),N(o)]],[discrete,[o,o],[M(o),N(o)]],[discrete,[o],[N(o)]]],[x,0,5],
[legend, "M(x)","N(x)", "Iteration_3","Iteration_4", "point"],
[style, [lines, 3,5], [lines, 3,1], [lines, 2,2],[lines, 5,3], [points,6,1 ]])$

Figure 5:

Iteration 4

(%i25)

i:o;

\[\]\[\tag{%o25} 2.006386175807663\]

(%i27)

m:M(i);o:X(m);

\[\]\[\tag{%o26} 1.805747558226897\]

\[\]\[\tag{%o27} 2.093521617375862\]

(%i28)

if (M(o)−N(o)<delta)then display("OK",M(o)−N(o))else display(M(o)−N(o));

\[\]\[\mbox{}\\"OK"\mathop{=}"OK" \]\[1.8841694556382758\mathop{+}\mathop{-}1.805747558226897\mathop{=}0.07842189741137884\]

\[\]\[\tag{%o28} \ensuremath{\mathrm{done}}\]

(%i29)

plot2d([M(x),N(x),[discrete,[i,o],[M(i),N(o)]],[discrete,[o,o],[M(o),N(o)]],[discrete,[o],[N(o)]]],[x,0,5],
[legend, "M(x)","N(x)", "Iteration_4","Iteration_5", "point"],
[style, [lines, 3,5], [lines, 3,1], [lines, 2,2],[lines, 5,3], [points,6,1 ]])$

Figure 6:

(%i30)

plot2d([M(x),N(x),[discrete,[i,o],[M(i),N(o)]],[discrete,[o,o],[M(o),N(o)]],[discrete,[o],[N(o)]]],[x,1.9,2.3],
[legend, "M(x)","N(x)", "Iteration_4","Iteration_5", "point"],
[style, [lines, 3,5], [lines, 3,1], [lines, 2,2],[lines, 5,3], [points,6,1 ]])$

(%i31)

del:M(o)−N(o);

\[\]\[\tag{%o31} 0.07842189741137884\]

Figure 7:

Created with wxMaxima.

The source of this Maxima session can be downloaded here.