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Thread: Using numerical methods to solve problems

  1. #1
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    Using numerical methods to solve problems

    Question One.
    Use an appropriate method to find the root of x + x + 1 = 0 correct to 2 decimal places. Start with initial interval -0.8 and -0.6. Show all working.

    Question Two.
    Human blood pressure P (in mm of mercury) varies periodically with time (t in seconds). The first time during the first second of a cycle when the pressure P is 90 mm can be expressed by:
    80 + 300t - 2700t = 90
    or:
    f(t) = 2700t - 300t + 10 = 0

    Take an initial value of t = 0.05 seconds and use a method different to Question One to find a solution to f(t) = 0 accurate to 4 decimal places.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Kiwigirl
    Question One.
    Use an appropriate method to find the root of x + x + 1 = 0 correct to 2 decimal places. Start with initial interval -0.8 and -0.6. Show all working.
    Let:

    $\displaystyle f(x)=x^3+x+1$.

    We wish to find a zero of $\displaystyle f$ in the interval $\displaystyle [-0.8,-0.6]$. We know it has a zero
    in this interval as it changes sign between $\displaystyle -0.8$ and $\displaystyle -0.6$.


    Here we are going to use binary chop, In the following tableau we have
    in each row a $\displaystyle x_1$ lower limit of the interval containing the zeros, $\displaystyle f_1$ the
    value of the function at the lower limit, $\displaystyle x_2$ the upper limit of the interval
    containing the zeros, $\displaystyle f_2$ the value of the function at the upper limit, $\displaystyle x_{mid}$
    the mid-point of the interval and $\displaystyle f_{mid}$ the value of the function at the mid=point.

    To obtain the following row from a row the value of the upper or lower limit
    of the interval is replaced by the mid-point according to the sign of the
    function at the mid-point. This process is continued until the upper and
    lower limits of the interval are equal to two significant digits.


    $\displaystyle \begin{array}{|c|c|c|c|c|c|}
    \hline x_1 & f_1 & x_2 & f_2 & x_{mid} & f_{mid}\\ \hline
    -0.8 & -0.312 & -0.6 & 0.184 & -0.7 & -0.043\\
    -0.7 & -0.043 & -0.6 & 0.184 & -0.65 & 0.074\\
    -0.7 & -0.043 & -0.65 & 0.074 & -0.675 & 0.175\\
    -0.7 & -0.043 & -0.675 & 0.175 & -0.6875 & -0.0124\\
    -0.6875 & -0.0124 &-0.675 & 0.175&-0.681&0.0026\\
    -0.6875 & -0.0124 &-0.681&0.0026&-0.684&-0.0046\\
    -0.684&-0.0046&-0.681&0.0026&\ & \ \\ \hline
    \end{array}$

    So the zero we seek is $\displaystyle x\approx -0.68$.

    RonL
    Last edited by CaptainBlack; May 5th 2006 at 10:53 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Kiwigirl
    Question Two.
    Human blood pressure P (in mm of mercury) varies periodically with time (t in seconds). The first time during the first second of a cycle when the pressure P is 90 mm can be expressed by:
    80 + 300t - 2700t = 90
    or:
    f(t) = 2700t - 300t + 10 = 0

    Take an initial value of t = 0.05 seconds and use a method different to Question One to find a solution to f(t) = 0 accurate to 4 decimal places.
    Here we will use Newton-Raphson itteration. This generates a new estimate
    for the root from an old estimate using the relation:

    $\displaystyle
    t_{new}=t_{old}-\frac{f(t_{old})}{f'(t_{old})}
    $


    $\displaystyle
    \begin{array}{|c|c|c|c|}\hline
    t_{old} & f(t_{old}) & f'(t_{old}) & t_{new} \\ \hline
    0.05 & -4.66 & -279.8 & 0.033333\\
    0.03333 & 0.10 & -291.0 & 0.033677\\
    0.033677 & 0.000025&-290.8 & 0.033677\\ \hline\end{array}
    $

    So to four significant digits $\displaystyle 0.033677$ is the required root.

    RonL
    Last edited by CaptainBlack; May 5th 2006 at 11:34 AM.
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