Roots of Equations

The Problem: Find x such that f(x)=0

Newton-Raphson Method

  • Guess
  • Improve guess as
  • Check convergence

Modified Newton-Raphson Method

  • Guess
  • Improve guess as
  • Check convergence

Test Cases

  1. Estimate the root of f(x) = e-x - x. (Ans. x = 0.5672)
  2. Estimate the non-zero root of f(x) = sin ( sqrt(x) ) - x. (Ans. x = 0.7686)
  3. Calculate the friction factor, f, from the Colbrook formula
    1/sqrt(f) = 2.0 log10 ( Re sqrt(f) ) - 0.8
    for Re = 20000, 400000, 2000000. (Ans. 0.02589, 0.01371, 0.01037)

Challenge Problems

  1. Estimate the smallest positive root of f(x) = - 0.0006x5 + 0.0205x4 - 0.2477x3 + 1.268x2 - 2.525x + 3.105.
  2. Estimate the root of f(x) = - 0.25 + ( 1/p )tan-1(x-3).

System of Non-linear Equations.

Search for a suitable routine for solving a system of equations in the GAMS website

  • We wish to solve a system of m equations



    in m variables

  • Guess
  • Solve the system of equations

  • Improve guess as
  • Check convergence

Download SNSQ for solving a system of non-linear equations.

Test Cases

  1. x2 + xy - 10 = 0
    y + 3xy2 - 57 = 0
    soln.: x = 2, y = 3
    second solution, courtesy of Chris Ambroson(Fall 2004): x = 4.3937, y= - 2.1178
  2. x = y + x2 - 0.5
    y = x2 - 5xy
    soln.: x = 1.233, y = 0.2122
    second solution, courtesy of Chris Ambroson(Fall 2004): x = - 0.1781, y= 0.2901
    third solution, courtesy of Chris Ambroson(Fall 2004): x = - 0.4552, y= - 0.1624