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Methods of iteration.
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Simple iteration.
AX = B, consider A = C+D, det C ¹ 0, then X = FX+G, F = C-1D, G = C-1B.
(1)
Let we find initial approximation:
X (0) = 
Next approximation we can find as X(k+1) = FX(k)+G (2)
Zeidel’s method of iteration.
On (k)-th step we use not only x(k–1), but x(k), which are already calculated.
Numerical solution of nonlinear systems.
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F1(x1, x2,..., xn)=0
F2(x1, x2,..., xn)=0
...........
Fn(x1, x2,..., xn)=0
Consider two-measured system:
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f1(x1, x2)=0 (1) Let we find bad approximation x1(0), x2(0)
f2(x1, x2)=0
Let’s try to find corrections:
x1=x1(0)+ d1; x2=x2(0)+ d2;
put them in (1):
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f1(x1(0)+ d1, x2(0)+ d2)=0
f2(x1(0)+ d1, x2(0)+ d2)=0
Decompose according to Tailor:
Consider only linear part:
(2)
We obtain system with variables d1, d2
So, we find first approximation:
x1(1)=x1(0)+ d1; x2(1)=x2(0)+ d2;
x1=x1(1)+ d1(1); x2=x2(1)+ d2(1);
and so on.
Numerical solution of differential equations
Ordinary differential equations.
Simplest form of O.D.E.
Analytic solution: 
.
Digital solution presents the table of function:
| x0 | x1 | x2 | … | xn | … |
| y0 | y1 | y2 | … | yn | … |
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