Secant method

In the secant method is a well-known since the Middle Ages numerical method for the approximate solution of an equation of the type. It corresponds to a simplification of the Newton's method, since not the derivative of the function to be calculated.

Method

Between two points and a secant function is defined. The intersection of a secant with the x- axis is used as an improved starting value for the iteration. Using the new function value is calculated. With the old value, and this step is repeated and a secant again. Repeating this step until a sufficient approximation of the desired zero was reached.

Construction on graph

The following animation shows how constructed with the starting values ​​and other points.

The procedure uses the following iteration:

It is started with two approximations.

In contrast to the Regula falsi method, it can occur in the secant method, that the neutral point for some iteration no longer between and is.

Related to the Newton's method

The method can be described as modification of the Newton approximation method with the iteration

Interpret, by the derivative of the difference quotient

Replaced.

Convergence

Because of the relationship to the Newtonian method similar conditions apply to the convergence of the secant method:

  • The secant method converges super linearly with the order of convergence, (this corresponds to the ratio of the golden section ), ie the number of correct digits of the approximation value increases approximately by a factor per pass. This is due to that the difference quotient is only an approximation of the derivative corresponding to lower the convergence speed in comparison with the square convergent Newton method.
  • The function must run continuously and have exactly one zero in the domain.
  • The method loses accuracy and convergence speed when the derivative at the root 0, since in the calculation results in an expression of the form. Especially with polynomials corresponds to a multiple zero.
  • In the numerical calculation, the problem arises that the difference quotient

Advantages of the process

Compared to the Newton's method results in several advantages:

  • It must be calculated only the function values ​​. In contrast to Newton iteration so that the zeros of any sufficiently smooth function can also be computed without the knowledge or calculation of the derivatives.
  • Each iteration has only the function f (x ) can be calculated once. When Newton's method must additionally also the function value of the derivative f '(x ) are determined.
  • By applying the two starting values ​​, the method can better focus on a specific interval, since the direction of the secant line is given by the two starting values ​​. The convergence can be characterized, however, not be forced.

The secant method in the multidimensional

Analogous to the multi-dimensional Newton's method can also define a multidimensional secant method to determine zeros of the functions.

Was in the one-dimensional derivative is approximated by the difference quotient, the Jacobian matrix is approximated in the multidimensional:

Which is defined at a given step size matrix by:

Now there is analogous to Newton iteration method comprising:

Since the dissolving

On the calculation of the inverse of a matrix and subsequent multiplication by complex and is numerically less favorable, is instead the system of linear equations

Solved. Then is obtained from:

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