Solution of Equations

Solution of Equations


  • The solution of equations involves finding values that satisfy a mathematical equation.
  • Equations can be linear (e.g. 2x + 3 = 7), quadratic (e.g. x^2 - 3x + 2 = 0), cubic (e.g. x^3 - 7x^2 + 14x - 8 = 0), polynomial (made of multiple terms, each term is a power of x) or transcendental (e.g. sin(x) = x^2).

Graphical Methods

  • Intersection of the graph of y=f(x) with y=0 gives the root of the equation f(x) = 0.
  • Sign change indicates a root between two values a and b if f(a) and f(b) have opposite signs.

Iterative Methods

  • Fixed point iteration involves rearranging the equation to x = g(x) and using an iterative formula x_n_+1 = g(x_n).
  • Convergence of iterative methods can be gauged using cobweb and staircase diagrams.

Newton-Raphson Method

  • The Newton-Raphson Method starts from an initial approximation and improves it by using the tangent line at the point of the function.
  • The iterative formula is x_n_+1 = x_n - f(x_n)/f’(x_n).
  • Tangent to the curve y=f(x) at a point P is obtained by differentiating f(x) to get f’(x).

Bisection Method

  • The bisection method involves selecting an interval [a, b] within which the root lies, and reducing the size of this interval by half at each step.
  • Check for a sign change within the interval. The root lies within the sub-interval where this occurs.
  • The midpoint (average of a and b) is taken as the next approximation for the root.

False Position Method (Regula-Falsi)

  • The false position method is similar to the bisection method, but instead of choosing the midpoint, a point on the line joining f(a) and f(b) is chosen.
  • This method may provide a more accurate approximation, particularly if the root is closer to one end of the interval.
  • The root’s approximation is given by the formula (af(b) - bf(a))/(f(b) - f(a)).

Convergence and Error Bound

  • Convergence refers to the tendency of an iterative method to reach the accurate root.
  • A method is convergent if the sequence of approximations tends to the root.
  • Error bound is the absolute difference between the root and the approximation. It can be used to judge the accuracy of the method.
  • Checking the rate of convergence can help you choose between methods for efficiency.

Rearranging Equations iteratively using the equation f(x)=0

  • Some equations can be rewritten in the form x = g(x), and you can then use iteration to find the root.
  • This involves rearranging the formula to make x the subject, substituting an initial value, and then continuing to substitute the result back into the formula until convergence occurs.
  • Be careful: not all equations will converge to a root using this method, and sometimes different rearrangements will lead to different roots.