ROOTS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS | NUMERICAL AND STATISTICAL METHODS

ROOTS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 



    Generally an equation is solved by factorization. 

    But in many cases, the method of factorization fails.

    In such cases, numerical methods are used.

    The numerical methods to solve the equation f(x) = 0 are:

    1) BISECTION METHOD 

    2) REGULA FALSI METHOD 

    3) NEWTON RAPHSON METHOD 

    4) SECANT METHOD



    1) BISECTION METHOD 

    Bisection Method is also called the interval halving method, the binary search method, or the dichotomy method.

    The Rate of Convergence of Bisection Method is Linear.

    For any continuous function f(x),

    Find two points say a and b such that f(a) < 0 and f(b) > 0

    Find midpoint of a and b say xn

    x= (a + b)/2

    Question :

    Obtain the root of the equation x – cosx = 0 by bisection method correct up to two decimal places. 

    Solution :

    f(x) = x – cosx

    f(0) = -1 and f(1) = 0.4597

    Since f(0) < 0 and f(1) > 0 , the root lies between 0 and 1.

    No.

    A

    B

    xn  = (a + b)/2

    Sign of f(xn)

    1

    0

    1

    0.5

    -ve

    2

    0.5

    1

    0.75

    +ve

    3

    0.5

    0.75

    0.625

    -ve

    4

    0.625

    0.75

    0.6875

    -ve

    5

    0.6875

    0.75

    0.71875

    -ve

    6

    0.71875

    0.75

    0.7344

    -ve

    7

    0.7344

    0.75

    0.7422

    +ve

    8

    0.7344

    0.7422

    0.7383

    -ve

    9

    0.7383

    0.7422

    0.74025

    +ve

    10

    0.7383

    0.74025

    0.7393

    +ve

    11

    0.7383

    0.7393

    0.7388

     

     Since x10 and x11 are same up to two decimal places, the root is 0.73  

    2) REGULA FALSI METHOD 

    Regula Falsi Method is also known as False Position Method.

    xn = [af(b) – bf(a)]/[f(b) – f(a)]

    Question:

    Find the real root of x log10 x – 1.2 = 0, correct up to three decimal places.

    Solution:

    f(x) = x log10 x – 1.2

    f(2) = -0.5979  and f(3) = 0.2314

    Since f(2) < 0 and f(3) > 0, the root lies between 2 and 3.

    No.

    a

    b

    f(a)

    f(b)

    xn =[af(b) – bf(a)]/[f(b) – f(a)]

    f(xn)

    1

    2

    3

    -0.5979

    0.2314

    2.721

    -0.0171

    2

    2.721

    3

    -0.0171

    0.2314

    2.7402

    -0.0004

    3

    2.7402

    3

    -0.0004

    0.2314

    2.7406

     

     The root is 2.740

    3) NEWTON-RAPHSON METHOD

    The Newton Raphson Method has a quadratic convergence and the convergence is of the order of 2.

    Newton raphson formula or Newton’s iteration formula is :

    xn+1 = xn – [f(xn)/f’(xn)]

    Question:

    Find  the real positive root of the equation x - cosx = 0 using Newton Raphson Method correct up to three decimal places.

    Solution:

    f(x) = x – cosx

    f(0) = -1 and f(1) = 0.4597

    Since f(0) < 0 and f(1) > 0 , the root lies between 0 and 1.

    Let x0 = 1

    f’(x) = 1 + sinx

    f(x0) = f(1) = 0.4597

    f(x0) = f’(1) = 1.8415

    x1 = x0 - [f(x0)/f’(x0)]

       = 1 – (0.4597/1.8415)

      = 0.7504

    f(x1) = f(0.7504) = 0.019

    f(x1) = f’(0.7504) = 1.6819

    x2 = x1 - [f(x1)/f’(x1)]

       = 0.7504 – (0.019/1.6819)

       = 0.7391

    f(x2) = f(0.7391) = 0.00002

    f(x2) = f’(0.7391) = 1.6736

    x3 = x2 - [f(x2)/f’(x2)]

       = 0.7391 – (0.00002/1.6736)

       = 0.7391

    Since x2 and x3 are same up to three decimal places, the root is 0.739

    4) SECANT METHOD 

    xn+1 = xn – {[(xn  - xn-1) * f(xn)]/ [f(xn) - f(xn-1)]}

    The rate of convergence of secant method is 1.618.

    The rate of convergence of secant method is lesser than Newton-Raphson Method and so Secant Method is more efficient than the Newton Raphson Method.

    Question:

    Find the root of xlog10 x – 1.9 = 0, correct up to three decimal places with x0 = 3 and x1 = 4.

    Solution:
    f(x) = xlog10 x – 1.9

    x0 = 3, x1 = 4

    f(x0) = f(3) = -0.4686, f(x1) = f(4) = 0.5082

    x2 = x1 – {[(x1  - x0) * f(x1)]/ [f(x1) - f(x0)]}

       = 4 – {[4 – 3) * (0.5082)]/[(0.5082 + 0.4686)]}

      = 3.4797

    f(x2) = f(3.4797) = -0.0156

    x3 = x2 – {[(x2 - x1) * f(x2)]/ [f(x2) - f(x1)]}

       = 3.4979 – {[3.4979 – 4) * (-0.0156)]/[(-0.0156 - 0.5082)]}

      = 3.4952

    f(x3) = f(3.4952) = -0.0005

    x4 = x3 – {[(x3  - x2) * f(x3)]/ [f(x3) - f(x2)]}

       = 3.4952 – {[3.4952 – 3.4797) * (-0.0005)]/[(-0.0005 + 0.01556)]}

       = 3.4957

    Since x3 and x4 are same up to three decimal places, the root is 3.495

    NOTES 

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