Numerical solution of equations.

F(x)=0 -equation;

if f(x) is polynomial, we have algebraic equation, in other case - transcendental.

The value c, wich transform equation to identity, named the root of equation: f(c)=0

EXAMPLE:

f(x)= x² -1; c=±1;

Precise solution can be found only in particular cases of equations.

But really it is not necessary.

We can find approximate value of root with any precision.

Primary value may be found by several ways, for example:

a) f(x)

x =c - graph crosses

abscissa

x

b) x x0 x1 x2... xn

f(x)=y y0 y1 y2... yn xk< c< xk+1,

if f(xa)*f(xk+1)< 0

In fact, we find some interval, contained c.

If some interval contains only one root, it responses such conditions:

cÎ [ a; b ]

1. f(a)*f(b)< 0

2. sign[f¢ (x)]=const, xÎ [a, b]

3. sign[f² (x)]=const, xÎ [a, b]

According to these conditions, we have four types of functions position:

1) 2)

f¢ > 0 f¢ > 0

f² > 0 f² < 0

		b		b

3 ) f¢ < 0 4) f¢ < 0

f² > 0 f² < 0

[a, b] named interval of isolation of root. The size of interval is error of root. We must reduce the size of interval of isolation.