Explanation Formula used: Newton’s method formula: p n + 1 = p n − f ( p n ) f ′ ( p n ) Calculation: Consider the amount of money required to pay off a mortgage over a fixed period of time involve the formula, A = P i [ 1 − ( 1 + i ) − n ] P i [ 1 − ( 1 + i ) − n ] − A = 0 P [ 1 − ( 1 + i ) − n ] − A × i = 0 Here A is the amount mortgage, P is the amount of each payment, and I is the interest rate per period for n payments. First write the function f ( i ) . f ( i ) = P [ 1 − ( 1 + i ) − n ] − A i Now, differentiate the above function with respect to i : f ′ ( i ) = P d d i [ 1 − ( 1 + i ) − n ] − d d i ( A i ) = P [ n ( 1 + i ) − ( n + 1 ) ] − A Use the Newton’s method, calculate the value of i k + 1 . i k + 1 = i k − f ( i k ) f ′ ( i k ) Substitute f ( i ) = P [ 1 − ( 1 + i ) − n ] − A i and f ′ ( i ) = P [ n ( 1 + i ) − ( n + 1 ) ] − A in i k + 1 = i k − f ( i k ) f ′ ( i k ) . i k + 1 = i k − P [ 1 − ( 1 + i k ) − n ] − A i k P [ n ( 1 + i k ) − ( n + 1 ) ] − A Since the payment is done monthly, the number of periods n should be 12 × 30 = 360 . Substitute A = $ 135 , 000 , P = $ 1 , 000 , and n = 360 in i k + 1 = i k − P [ 1 − ( 1 + i k ) − n ] − A i k P [ n ( 1 + i k ) − ( n + 1 ) ] − A . i k + 1 = i k − 1 , 000 [ 1 − ( 1 + i k ) − 360 ] − 135 , 000 i k 1 , 000 [ 360 ( 1 + i k ) − 361 ] − 135 , 000 The solution of the above equation can be calculated using Newton’s method formula. The value of I lies between the interval [ 0 , 1 ] and start with the initial approximation i 0 = 0.5 . Now, calculate the first iteration by substituting k = 0 and i 0 = 0.5 in i k + 1 = i k − 1 , 000 [ 1 − ( 1 + i k ) − 360 ] − 135 , 000 i k 1 , 000 [ 360 ( 1 + i k ) − 361 ] − 135 , 000 . i 1 = i 0 − 1 , 000 [ 1 − ( 1 + i 0 ) − 360 ] − 135 , 000 i 0 1 , 000 [ 360 ( 1 + i 0 ) − 361 ] − 135 , 000 = 0.5 − 1 , 000 [ 1 − ( 1 + 0.5 ) − 360 ] − 135 , 000 ( 0.5 ) 1 , 000 [ 360 ( 1 + 0.5 ) − 361 ] − 135 , 000 = 0.00740740 Now, calculate the second iteration by substituting k = 1 and i 1 = 0.00740740 in i k + 1 = i k − 1 , 000 [ 1 − ( 1 + i k ) − 360 ] − 135 , 000 i k 1 , 000 [ 360 ( 1 + i k ) − 361 ] − 135 , 000

10th Edition

ISBN: 9781305253667

Author: Richard L. Burden, J. Douglas Faires, Annette M. Burden

Publisher: Cengage Learning

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Chapter 2.3, Problem 23ES

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Chapter 2 Solutions

Numerical Analysis

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Chapter 2 Solutions