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    amarvenkat30
    This document is a homework assignment for a computer simulation course. It contains two exercises. The first asks students to derive forward and backward iterations for integrals involving natural logarithms, determine which is stable, and write a MATLAB program to calculate the first 20 terms. The second exercise involves analyzing the iterative solution to the mean-field Ising model equation near a critical point, deriving the leading-order solution, determining convergence rates, and writing a MATLAB program to plot the error over iterations. Students are instructed to show their work and explanations for any incomplete programs.

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    Sheet 2

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    amarvenkat30
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    This document is a homework assignment for a computer simulation course. It contains two exercises. The first asks students to derive forward and backward iterations for integrals involving natural logarithms, determine which is stable, and write a MATLAB program to calculate the first 20 terms. The second exercise involves analyzing the iterative solution to the mean-field Ising model equation near a critical point, deriving the leading-order solution, determining convergence rates, and writing a MATLAB program to plot the error over iterations. Students are instructed to show their work and explanations for any incomplete programs.

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    This document is a homework assignment for a computer simulation course. It contains two exercises. The first asks students to derive forward and backward iterations for integrals involving natural logarithms, determine which is stable, and write a MATLAB program to calculate the first 20 terms. The second exercise involves analyzing the iterative solution to the mean-field Ising model equation near a critical point, deriving the leading-order solution, determining convergence rates, and writing a MATLAB program to plot the error over iterations. Students are instructed to show their work and explanations for any incomplete programs.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    1 views2 pages

    Sheet 2

    Uploaded by

    amarvenkat30
    This document is a homework assignment for a computer simulation course. It contains two exercises. The first asks students to derive forward and backward iterations for integrals involving natural logarithms, determine which is stable, and write a MATLAB program to calculate the first 20 terms. The second exercise involves analyzing the iterative solution to the mean-field Ising model equation near a critical point, deriving the leading-order solution, determining convergence rates, and writing a MATLAB program to plot the error over iterations. Students are instructed to show their work and explanations for any incomplete programs.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    CSIS

    Computer Simulation in Science

    Introduction to Computer Simulation


    WS 23/24
    Prof. Dr. F. Knechtli
    M.Sc. T. Asmussen, Dr. J. A. Urrea-Niño

    Homework 2
    due 06.11.23 before 8 AM
    Solutions must be delivered as specified by the instructions shown in Moodle.

    Exercise 2.1:

    Consider the integrals Z e


    Pn = [ln(x)]n dx. (1)
    1

    Derive a forward iteration Pn+1 = f (Pn ) and a backward iteration Pn = g (Pn+1 ).


    Determine which one of the two is stable. These previous calculations must be
    done by hand. Finally, write a Matlab program which uses the stable iteration to
    calculate the first 20 terms P1 , P2 , ..., P20 and prints them to the command line.

    (10 points)

    Exercise 2.2:
    The mean-field equation for the Ising model is

    x⋆ = tanh(6βx⋆ ) . (2)

    The iterative solution of Eq. (2) builds up a sequence

    xn+1 = tanh(6βxn ) , x0 > 0 , (3)

    which converges to x⋆ . We analyze the case where β is slightly larger than 1/6:

    6β = 1 + ϵ with ϵ ≪ 1

    and the non-trivial solution x⋆ ̸= 0 of Eq. (2) becomes very “small”.

    a) Expand the solution x⋆ to leading order in ϵ [hint: Taylor expansion of


    Eq. (2)]. Then discuss the convergence of Eq. (3): to this end write xn =
    x⋆ + δn (and similarly for xn+1 ) and expand Eq. (3) to leading order in the
    errors δ. How many iterations N are needed to reduce the error by a factor
    of 10? The computations must be done by hand.

    1
    b) Write a Matlab program which calculates the estimates xn and plots the
    error δn as a function of n on a semilogarithmic scale. Set β = 0.17 and
    in this case x⋆ = 0.24062696159732. Does the error behave as expected?
    How many iterations are needed to obtain the solution to machine accuracy?
    Compare this number with the result presented in the lecture. Compare
    the number of iterations with the analytic result obtained in part a) of the
    exercise. What can cause differences between these two values?

    (15 points)
    Note: For each exercise we should be able to see, from what you handed in, what
    you did and what came out of it. Print out/write down everything we need for
    this. If you don’t manage to write a working program, then write down a kind of
    pseudo code. That is, explain step by step, what your program should be doing.

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