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POTaylor Series Calculating Program in C compiled with GCC
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copied!<p>I have written a program in C that calculates the approximate value of the function exp(x) at various values of x, some positive and some negative. My questions are as follows:</p> <ol> <li>Even though I was able to include 151 terms in the sequence before, now the series blows up for exp(100) even at the 140th term. Afterwards I messed up more with the code and now the series blows up even at the 130 term. In addition to that I have used long double as the number format to allow large factorials to be calculated and displayed. How can one explain this phenomenon? Sometimes after successive execution of the code I get results approximating the real value of the function which I checked from Wolfram Alpha.</li> <li>I have heard that "for" loops proved to be inefficient; do you have alternatives for "for" loop in mind? I would be happy to try to implement them as much as I can.</li> <li>I want to display the output in scientific format but the format specifier (I am not sure about the term) did not allow me to do so. The specifier I used in printf statement was %g. How can I have the output displayed in scientific format?</li> <li>Based on the hardware and software I am using is this the maximum precision I can get?</li> <li>I also getting a Segmentation Fault (core dumped) error instead of the display for x = -100. What may be the reason behind this?</li> </ol> <p>Thanks for your contribution. I appreciate any help and recommendations. </p> <p>P.S: I compile using GCC on 64-bit hardware, but 32-bit Ubuntu 12.04 LTS (a.k.a Precise Pangolin).</p> <pre><code>#include <stdio.h> #include <math.h> //We need to write a factorial function beforehand, since we //have factorial in the denominators. //Remembering that factorials are defined for integers; it is //possible to define factorials of non-integer numbers using //Gamma Function but we will omit that. //We first declare the factorial function as follows: long double factorial (double); //Long long integer format only allows numbers in the order of 10^18 so //we shall use the sign bit in order to increase our range. //Now we define it, long double factorial(double n) { //Here s is the free parameter which is increased by one in each step and //pro is the initial product and by setting pro to be 0 we also cover the //case of zero factorial. int s = 1; long double pro = 1; if (n < 0) printf("Factorial is not defined for a negative number \n"); else { while (n >= s) { pro *= s; s++; } return pro; } } int main () { long double x[13] = { 1, 5, 10, 15, 20, 50, 100, -1, -5, -10, -20, -50, -100}; //Here an array named "calc" is defined to store //the values of x. //The upper index controls the accuracy of the Taylor Series, so //it is suitable to make it an adjustable parameter. However, after //a certain value of p the series become infinitely large to be //represented by the computer; hence we terminate the series at //the 151th term. int p = 150; long double series[13][p]; int i, k; //We only define the Taylor series for positive values of the exponent, and //later we will use these values to calculate the reciprocals. This is done //in this manner to avoid the ambiguity introduced into the sum due to terms //alternating in sign. long double sum[6] = { 0 }; for (i = 0; i <= 6;i++) { for (k = 0; k <= p; k++){ series[i][k] = pow(x[i], k)/( factorial(k)); sum[i] += series[i][k]; } printf("Approximation for x = %Lf is %Lf \n", x[i], sum[i]); } //For negative numbers -taking into account first negative number is //in the 8th position in the x array and still denoting the approximation long double approx[5] = { 0 }; for (i = 7; i <= 12;i++) { approx[i - 7] = 1 / sum[i - 7]; printf("Approximation for x = %Lf is %Lf \n", x[i], approx[i - 7]); } //printf("%Lf \n", factorial(3)); //The above line was introduced to test if the factorial function //was functioning properly. } </code></pre>

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