Secant for particular equation; Program to read a Non-Linear equation in one variable. Fortran Numerical Analysis Programs. Write a Fortran program to. Fortran Numerical Analysis Programs. = 3 x + sin x − e x using Secant method in the interval. Write a Fortran program to find first derivation of the.: Orthogonal polynomials generator.
Bisection method is used to find the real roots of a nonlinear equation. The process is based on the ‘Intermediate Value Theorem‘. According to the theorem “If a function f(x)=0 is continuous in an interval (a,b), such that f(a) and f(b) are of opposite nature or opposite signs, then there exists at least one or an odd number of roots between a and b.”
In this post, the algorithm and flowchart for bisection method has been presented along with its salient features.
Bisection method is a closed bracket method and requires two initial guesses. It is the simplest method with slow but steady rate of convergence. It never fails! The overall accuracy obtained is very good, so it is more reliable in comparison to the Regula-Falsi method or the Newton-Raphson method.
Features of Bisection Method:
- Type – closed bracket
- No. of initial guesses – 2
- Convergence – linear
- Rate of convergence – slow but steady
- Accuracy – good
- Programming effort – easy
- Approach – middle point
Bisection Method Algorithm:
- Start
- Read x1, x2, e
*Here x1 and x2 are initial guesses
e is the absolute error i.e. the desired degree of accuracy* - Compute: f1 = f(x1) and f2 = f(x2)
- If (f1*f2) > 0, then display initial guesses are wrong and goto (11).
Otherwise continue. - x = (x1 + x2)/2
- If ( [ (x1 – x2)/x ] < e ), then display x and goto (11).
* Here [ ] refers to the modulus sign. * - Else, f = f(x)
- If ((f*f1) > 0), then x1 = x and f1 = f.
- Else, x2 = x and f2 = f.
- Goto (5).
*Now the loop continues with new values.* - Stop
Bisection Method Flowchart:
The algorithm and flowchart presented above can be used to understand how bisection method works and to write program for bisection method in any programming language.
Also see,
Bisection Method C Program
Bisection Method MATLAB Program
Bisection Method C Program
Bisection Method MATLAB Program
Note: Bisection method guarantees the convergence of a function f(x) if it is continuous on the interval [a,b] (denoted by x1 and x2 in the above algorithm. For this, f(a) and f(b) should be of opposite nature i.e. opposite signs.
The slow convergence in bisection method is due to the fact that the absolute error is halved at each step. Due to this the method undergoes linear convergence, which is comparatively slower than the Newton-Raphson method, Secant method and False Position method.
Secant method is considered to be the most effective approach to find the root of a non-linear function. It is a generalized from the Newton-Raphson method and does not require obtaining the derivatives of the function. So, this method is generally used as an alternative to Newton Raphson method.
Secant method falls under open bracket type. The programming effort may be a tedious to some extent, but the secant method algorithm and flowchart is easy to understand and use for coding in any high level programming language.
This method uses two initial guesses and finds the root of a function through interpolation approach. Here, at each successive iteration, two of the most recent guesses are used. That means, two most recent fresh values are used to find out the next approximation.
Features of Secant Method:
- No. of initial guesses – 2
- Type – open bracket
- Rate of convergence – faster
- Convergence – super linear
- Accuracy – good
- Approach – interpolation
- Programming effort – tedious
Secant Method Algorithm:
- Start
- Get values of x0, x1 and e
*Here x0 and x1 are the two initial guesses
e is the stopping criteria, absolute error or the desired degree of accuracy* - Compute f(x0) and f(x1)
- Compute x2 = [x0*f(x1) – x1*f(x0)] / [f(x1) – f(x0)]
- Test for accuracy of x2
If [ (x2 – x1)/x2 ] > e, *Here [ ] is used as modulus sign*
then assign x0 = x1 and x1 = x2
goto step 4
Else,
goto step 6 - Display the required root as x2.
- Stop
Secant Method Flowchart:
Also see,
Secant Method C Program
Secant Method MATLAB Program
Secant Method C Program
Secant Method MATLAB Program
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Secant method is an improvement over the Regula-Falsi method, as successive approximations are done using a secant line passing through the points during each iteration. Following the secant method algorithm and flowchart given above, it is not compulsory that the approximated interval should contain the root.
Secant method is faster than other numerical methods, except the Newton Raphson method. Its rate of convergence is 1.62, which is quite fast and high. However, convergence is not always guaranteed in this method. But, overall, this method proves to be the most economical one to find the root of a function.