7 root finding in one dimension
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7 Root Finding in One DimensionTRANSCRIPT
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