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(2) (a) Refer to the course notes on Algorithm Stability and note the definition for En· Using an integration by parts approach we were able to show that, En = 1 nEn-1. Once we were able to figure out a value for E₁, the above recurrence formula suggested a straightforward way to obtain whatever En values we wanted, for n > 1. However, the numerical results showed that something went terribly wrong with the computation. Explain in your own words why the computation goes terribly wrong. (b) In the course notes, we saw that a simple rewriting of the recurrence formula, and a fairly rough approximation for En, for a larger value of n, leads to an algorithm which provides very accurate results for smaller values of n. Explain in your own words why this approach work so well. (c) Implement the two algorithms in Fortran to reproduce the numerical results shown in the notes. Verify that you get the same results. (Start by editing the Stirling Approx- imation program for n! from Assignment #1.)

(2) (a) Refer to the course notes on Algorithm Stability and note the definition for En· Using an integration by parts approach we were able to show that, En = 1 nEn-1. Once we were able to figure out a value for E₁, the above recurrence formula suggested a straightforward way to obtain whatever En values we wanted, for n > 1. However, the numerical results showed that something went terribly wrong with the computation. Explain in your own words why the computation goes terribly wrong. (b) In the course notes, we saw that a simple rewriting of the recurrence formula, and a fairly rough approximation for En, for a larger value of n, leads to an algorithm which provides very accurate results for smaller values of n. Explain in your own words why this approach work so well. (c) Implement the two algorithms in Fortran to reproduce the numerical results shown in the notes. Verify that you get the same results. (Start by editing the Stirling Approx- imation program for n! from Assignment #1.)

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter14: Numerical Methods
Section14.3: Refinements To The Bisection Method
Problem 4E
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(2) (a) Refer to the course notes on Algorithm Stability and note the definition for En· Using an integration by parts approach we were able to show that, En = 1 nEn-1. Once we were able to figure out a value for E₁, the above recurrence formula suggested a straightforward way to obtain whatever En values we wanted, for n > 1. However, the numerical results showed that something went terribly wrong with the computation. Explain in your own words why the computation goes terribly wrong. (b) In the course notes, we saw that a simple rewriting of the recurrence formula, and a fairly rough approximation for En, for a larger value of n, leads to an algorithm which provides very accurate results for smaller values of n. Explain in your own words why this approach work so well. (c) Implement the two algorithms in Fortran to reproduce the numerical results shown in the notes. Verify that you get the same results. (Start by editing the Stirling Approx- imation program for n! from Assignment #1.)
Transcribed Image Text:(2) (a) Refer to the course notes on Algorithm Stability and note the definition for En· Using an integration by parts approach we were able to show that, En = 1 nEn-1. Once we were able to figure out a value for E₁, the above recurrence formula suggested a straightforward way to obtain whatever En values we wanted, for n > 1. However, the numerical results showed that something went terribly wrong with the computation. Explain in your own words why the computation goes terribly wrong. (b) In the course notes, we saw that a simple rewriting of the recurrence formula, and a fairly rough approximation for En, for a larger value of n, leads to an algorithm which provides very accurate results for smaller values of n. Explain in your own words why this approach work so well. (c) Implement the two algorithms in Fortran to reproduce the numerical results shown in the notes. Verify that you get the same results. (Start by editing the Stirling Approx- imation program for n! from Assignment #1.)
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