Method of the secant

In numerical Analysis, the method of the secant is an algorithm of search for roots of a function F .

Method

The method of the secant is a method derived from that of Newton where one replaces $f\text{'}\; (x\_n)\; \backslash ,$ by $\backslash \; frac\; \{F\; (x\_n)\; -\; F\; (x\_\; \{n-1\})\}\{x\_n-x\_\{n-1\}\}$ One obtains the relation of Récurrence

$x\_\; \{n+1\}\; =\; x\_n\; -\; \backslash \; frac\; \{x\_n-x\_\; \{n-1\}\}\; \{F\; (x\_n)\; -\; F\; (x\_\; \{n-1\})\}\; F\; (x\_n).$

Initialization requires 2 points X ₀ and X ₁, close relations, if possible, of the required solution.

Demonstration

Being given has and B , one builds the segment connecting ( has , F ( has )) and ( B , F ( B )). The line can be as follows defined:

$there\; -\; F\; (b)\; =\; \backslash \; frac\; \{F\; (B)\; -\; F\; (a)\}\; \{Ba\}\; (x-b).$

C is chosen so that C is the root of this line (i.e., F ( C ) =0).

$F\; (b)\; +\; \backslash \; frac\; \{F\; (B)\; -\; F\; (a)\}\; \{Ba\}\; (c-b)\; =\; 0.$

If one extracts C from this equation, one finds the relation of recurrence referred to above.

Convergence

If the initial values of X ₀ and X ₁ are sufficiently close to the solution, the method will have a Ordre of convergence of

$\backslash \; varphi\; =\; \backslash \; frac\; \{1+\; \backslash \; sqrt\; \{5\}\}\; \{2\}\; \backslash \; simeq\; 1,618$ which is the Golden section.

However, the function F must be 2 times continuement differentiable and it must be a simple root.

Example of implementation

This program in C solves the problem F ( X ) = cos ( X ) - X ³ = 0. The tests of stops are the following:

• $N\; >\; m$

m and E being given.

# include # include   double F ( double X) { return cos (X) - x*x*x; }   double SecantMethod ( double xn_1, double xn, double E, int m) { int N; double D; for (N = 1; N <= m; n++) { D = (xn - xn_1)/(F (xn) - F (xn_1)) * F (xn); yew (fabs (d) < E) return xn; xn_1 = xn; xn = xn - D; } return xn; }   int hand ( void ) { printf (" %0.15f \ n" , SecantMethod (0, 1,5E-11, 100)); return 0; }

The following results are obtained:

$x\_0\; =\; 0\; \backslash ,\; \backslash !$

$x\_1\; =\; 1\; \backslash ,\; \backslash !$

$x\_2\; =\; 0,685073357326045\; \backslash ,\; \backslash !$

$x\_3\; =\; 0,841355125665652\; \backslash ,\; \backslash !$

$x\_4\; =\; 0,870353470875526\; \backslash ,\; \backslash !$

$x\_5\; =\; 0,865358300319342\; \backslash ,\; \backslash !$

$x\_6\; =\; 0,865473486654304\; \backslash ,\; \backslash !$

$x\_7\; =\; 0,865474033163012\; \backslash ,\; \backslash !$

$x\_8\; =\; 0,865474033101614\; \backslash ,\; \backslash !$

See too

• Method of Newton
• Method of the false position

Random links:

Paul Cohen | Suger of Saint-Denis | Something Else by the Kinks | William Witney | Tomuya | _de_Clementina_Sobieski

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