Root finding in one dimension

non-linear equation (Root finding)

23-Aug-152Mr.N Kannan, MIT, Manipal

Bisection methodThis is the simplest method for finding a root to an equation. We need lower and higher (xa and xb) values which bracket the root: let fa = f(xa) and fb = f(xb) such that fa fb 0 then the root lies in the interval (xa,xc),if fc < 0 then the root lies in the interval (xc,xb),if fc < 0 then the root lies in the interval (xc,xb),abalgorithmThe basic algorithm for the bisection method,Let xc = (xa+xb)/2,if fc = f(c) = 0 then x = xc is an exact solution,elseif fa fc < 0 then the root lies in the interval (xa,xc),else the root lies in the interval (xc,xb).By replacing the interval (xa,xb) with either (xa,xc) or (xc,xb) (whichever brackets the root), the error in our estimate of the solution to f(x) = 0. We repeat this interval halving until either the exact root has been found or the interval is smaller than some specified tolerance.23-Aug-156Mr.N Kannan, MIT, Manipal23-Aug-157Mr.N Kannan, MIT, Manipal

INPUT VALUES23-Aug-158Mr.N Kannan, MIT, Manipal

23-Aug-159Mr.N Kannan, MIT, ManipalLinear interpolation (regula falsi)This method is similar to the bisection method in that it requires two initial guesses to bracket the root. Instead of simply dividing the region in two, a linear interpolation is used to obtain a new point which is (hopefully, but not necessarily) closer to the root than the equivalent estimate for the bisection method. 23-Aug-1510Mr.N Kannan, MIT, Manipal

Regula Falsi MethodBy the principle of slope of a line we have

Then c will replace a according to Figure, since f(c) < O.

algorithm

23-Aug-1512Mr.N Kannan, MIT, Manipal23-Aug-15Mr.N Kannan, MIT, Manipal13

23-Aug-15Mr.N Kannan, MIT, Manipal14

Example :The heat capacity of carbon dioxide is given as a function of temperature as

where the units of Cp are (kJ/kg K) and the unit of temperature T is (K). determine the temperature which yields a value of the heat capacity of 1 (kJ/kg K).Initial GuessT = 400 K and T = 600 K

IterationabF( a )F ( b )cF ( c )1400600-0.0380.0995000.0412400500-0.0380.0414500.0051 3400450-0.0380.0051425-0.0154425450-0.0150.0051437.5-0.00495437.5450-0.00490.0051443.750.000146437.5443.75-0.00490.00014440.625-0.00247440.625443.75-0.00240.00014

The convergence using Regular Falsi is more rapid than that by bisection.

Regula Falsi Method :18Newton Raphson Method -This widely used derivative-based method for a single nonlinear algebraic equation. - Here we solve equation f (x) = 0.

The slope or derivative of f(x) is computed at an initial guess, a, for the root of f(x) = 0 . The new value of the root, b, is computed based on a first-order Taylor series expansion of f(x) about the initial guess, a.

This method is iterative, but it only requires one initial guess

An important advantage of the Newton method is its rapid convergence.

Algorithm of Newton Raphson MethodStep 1: Selection of initial approximation in values of x i.e. x0.Step 2: Find the value of f(x) & f(x) at x0Step 3: Find the new approximation xi by Newton Raphson method i.e. xi = x0 f(x0)/f(x0)Step 4: Now replace the value of x0 by xi.Step 5: Repeat step 2: through to step 4: until the absolute value off(x) at xi becomes equal to zero (or less than tolerance).Step 6: Hence the required root is xi.

23-Aug-15Mr.N Kannan, MIT, Manipal23

23-Aug-15Mr.N Kannan, MIT, Manipal24