Exponential Taylor Series - c

This is the code I have so far, which is a little messy since I am still trying to figure out how to set it up, but I cannot figure out how to get the output. This code is supposed to take a Taylor Series polynomial of an exponential, and check the amount of iterations it takes to get the approximation.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
/*Prototype for functions used*/
double factorial (int);
int main()
{
double input = 0;
double exp_val;
double delta = 1;
int f =0;
int n = 0;
double taylor;
int total;
printf("Plese enter the exponent to check for convergence:\n");
scanf("%lf", &input);
exp_val = exp(input);
printf(" # Iter e^X Sum Diff\n");
printf("---- ------ ------- ----- --------");
while(delta > 0.00001)
{
f = factorial(n);
taylor = ((pow(input,n))/ f);
delta = (exp_val - taylor);
printf("%d %f %f %f/n", (n+1), exp_val, taylor, delta);
n++;
}
system("pause");
}
double factorial (int n)
{
int r = 0;
int sum = 1;
int total = 0;
if (n == 0)
return total =1;
else
{
for(r; r<n; r++)
{
sum = sum * r;
total = sum + 1;
}
return total;
}
}

Here, I have fixed it, without changing your approach, except for the parts I really had to. One thing we have to clarify before the code is how Taylor Polynomials are made. It is not the first term plus the nth term, rather the sum of all terms from the first term till the nth term. So you definitely have to increase the taylor variable by the current nth term instead of the other way.
Here's the code, with brief comments in it as the explanation:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
/*Prototype for functions used*/
unsigned long long factorial( int ); // <-- made it return unsigned long long
int main( )
{
double input = 0;
double exp_val;
double delta = 1;
unsigned long long f = 0; // <-- changed its type
int n = 0;
double taylor = 0; // <-- initialized with 0
printf( "Plese enter the exponent to check for convergence:\n" );
scanf( "%lf", &input );
exp_val = exp( input );
printf( " # e^X Sum Diff\n" ); // <-- made some cosmetic changes
printf( "--- --------- --------- ---------\n" ); // <-- added \n
while ( delta > 0.00001 )
{
f = factorial( n );
taylor += ( ( pow( input, n ) ) / f ); // += instead of =
delta = ( exp_val - taylor );
printf( "%2d %12f %12f %12f\n", ( n + 1 ), exp_val, taylor, delta ); // <-- replaced / with \ before the n
n++; // and made some edits to make it look better
}
system( "pause" );
return 0; // <-- better add this
}
unsigned long long factorial( int n ) // <-- made it return unsigned long long
{
int r = 0;
unsigned long long sum = 1; // <-- changed its type
if ( n == 0 )
return sum; // <-- this
else
{
for ( r; r<n; r++ )
{
sum *= r + 1; // <-- changed this
}
return sum; // <-- and this
}
}
You have to keep in mind that you may not input too high values to it. Anything higher than input == 4 kind of breaks it, because, you see, even with 4, it can reduce the error delta beneath the threshold first only with the 19th cycle. The programme seemingly fails with n == 5 due to inaccurate calculation of pow( 5, 21 ) / factorial( 21 ) when n reaches 21:
0.000034 // the result this programme finds
0.0000093331055943447405008542892329719 // the result Calculator finds
So, yeah... If you want this programme to work with bigger input values, you'll need a better approach. Not calculating the nth term from scratch and calculating it from the (n - 1)th term instead could help until somewhat bigger input values, as the others had said.

A couple issue:
Change int r = 0; ... for(r; r<n; r++) to int r; ... for(r=1; r<=n; r++) or int r = 1; ... for(; r<=n; r++)
Change printf("%d %f %f %f/n" to printf("%d %f %f %f\n" Add \n
Change "... --------" to "... --------\n"
Change delta = (exp_val - taylor); to delta = fabs(exp_val - taylor);
Change to double taylor = 0.0; Initialize it.
Change to taylor += ((pow(input,n))/ f); Note: +=
Minor: "Please" not "Plese".
Minor: Drop int total;

Related

Trapezoidal Integration in C

I am trying to compute the integral of the function f(x)=(1-x^2)^(1/2) from x=0 to x=1. The answer should be approximately pi/4. I am currently getting 2.
My current implementation of the trapezoidal rule is the following:
double
def_integral(double *f, double *x, int n)
{
double F;
for (int i = 0 ; i < n ; i++) {
F += 0.5 * ( x[i+1] - x[i] ) * ( f[i] + f[i+1] );
}
return F;
}
I'm creating N divisions to approximate the area under the curve between x_1=0 and x_N=1 by looping through i to N with x_i = i / N.
int
main(int argc, char **argv)
{
int N = 1000;
double f_x[N];
double x[N];
for (int i = 0 ; i <= N ; i++) {
double x = i * 1. / N;
f_x[i] = sqrt(1. - pow(x, 2.));
//printf("%.2f %.5f\n", x, f_x[i]); //uncomment if you wanna see function values
}
double F_x = def_integral(f_x, x, N);
printf("The integral is %g\n", F_x);
}
The result of 2 that I am currently getting should be dependent on the number of N division, however, no matter if I make N=10000 or N=100, I still get 2.
Any suggestions?
In this for loop, you forgot updatin array x as well.
for (int i = 0 ; i <= N ; i++) {
double x = i * 1. / N;
f_x[i] = sqrt(1. - pow(x, 2.));
//printf("%.2f %.5f\n", x, f_x[i]); //uncomment if you wanna see function values
}
So, for loop should be replaced by
for (int i = 0 ; i <= N ; i++) {
double xi = i * 1. / N;
x[i] = xi;
f_x[i] = sqrt(1. - pow(xi , 2.));
//printf("%.2f %.5f\n", x, f_x[i]); //uncomment if you wanna see function values
}
In your main code, you call def_integral with a double (x) and in the function an array of x (double * x) is expected. Perhaps (it is what I suppose), the problem comes from the fact you formula needs x(i+1)-x(i) but you use a constant step. Indeed, x(i+1)-x(i)=step_x is constant so you do not need each x(i) but only value : 1./N
Other remark, with a constant step, your formula could be simplified to : F_x=step_x* ( 0.5*f_x(x0)+ f_x(x1)+...+f_x(xn-1)+ 0.5*f_x(xn) ) . It helps to simplify the code and to write a better efficient one.
Everything is commented in the code above. I hope it could help you. Best regards.
#include <stdio.h>
#include <math.h>
double
def_integral(double *f, double step_x, int n)
{
double F;
for (int i = 0 ; i < n ; i++) {
F += 0.5 * ( step_x ) * ( f[i] + f[i+1] );
}
return F;
}
int main()
{
int N = 1001; // 1001 abscissas means 1000 intervalls (see comment on array size and indices)
double f_x[N]; // not needed for the simplified algorithm
double step_x = 1. / N; // x(i+1)-x(i) is constant
for (int i = 0 ; i < N ; i++) { // Note : i<N and not i<=N
double xi = i * step_x; // abscissa calculation
f_x[i] = sqrt((1. - xi )*(1. + xi )); // cf chux comment
}
double F_x = def_integral(f_x, step_x, N);
printf("The integral is %.10g\n", F_x);
// simplified algorithm
// F_x=step_x*( 0.5*f_x(x0)+f_x(x1)+...+f_x(xn-1)+0.5f_x(xn) )
double xi;
xi=0; // x(0)
F_x=0.5*sqrt((1. - xi )*(1. + xi ));
for (int i=1 ; i<=N-1 ; i++) {
xi=step_x*i;
F_x+=sqrt((1. - xi )*(1. + xi ));
}
xi=step_x*N;
F_x+=0.5*sqrt((1. - xi )*(1. + xi ));
F_x=step_x*F_x;
printf("The integral is %.10g\n", F_x);
}

What should I change so that my arctan(x) approximation can display x=1 and x=-1 properly?

One of my C assignments was it to write an approximation of arctan(x) in the language C. The equation which I should base it on is
arctan(x)=\sum {k=0}^{\infty }(-1)^{k} \tfrac{x^{2k+1}}{2k+1}
In addition x is only defined as -1<=x<=1.
Here is my code.
#include <stdio.h>
#include <math.h>
double main(void) {
double x=1;
double k;
double sum;
double sum_old;
int count;
double pw(double y, double n) {
double i;
double number = 1;
for (i = 0; i < n; i++) {
number *= y;
}
return(number);
}
double fc (double y) {
double i;
double number = 1;
for (i = 1; i <= y; i++){
number *= i;
}
return(number);
}
if(x >= (-1) && x <= 1) {
for(k=0; sum!=sum_old; k++) {
sum_old = sum;
sum += pw((-1), k) * pw(x, (2*k) + 1)/((2*k) + 1);
count++;
printf("%d || %.17lf\n", count, sum);
}
printf("My result is: %.17lf\n",sum);
printf("atan(%f) is: %.17f\n", x, atan(x));
printf("My result minus atan(x) = %.17lf\n", sum - atan(x));
} else {
printf("x is not defined. Please choose an x in the intervall [-1, 1]\n");
}
return 0;
}
It seemingly works fine with every value, except value 1 and -1. If x=1, then the output ends with:
...
7207 || 0.78543285189457468
7208 || 0.78536
Whereas the output should look more like this. In this case x=0.5.
25 || 0.46364760900080587
26 || 0.46364760900080587
My result is: 0.46364760900080587
atan(0.500000) is: 0.46364760900080609
My result minus atan(x) atan(x) = -0.00000000000000022
How can I improve my code so that it can run with x=1 and x=-1.
Thanks in advance.
PS: I use my own created pw() function instead of pow(), because I wanted to bybass the restriction of not using pow() as we didn't had that in our lectures yet.
PPS: I'd appreciate any advice as to how to improve my code.
In each iteration, you add (-1)k • x2k+1 / (2k+1), and you stop when there is no change to the sum.
If this were calculated with ideal arithmetic (exact, infinitely precise arithmetic), it would never stop for non-zero x, since you are always changing the sum. When calculating with fixed-precision arithmetic, it stops when the term is so small it does not change the sum because of the limited precision.
When |x| is less than one by any significant amount, this comes quickly because x2k+1 gets smaller. When |x| is one, the term becomes just 1 / (2k+1), which gets smaller very slowly. Not until k is around 253 would the sum stop changing.
You might consider changing your stopping condition to be when sum has not changed from sum_old very much rather than when it has not changed at all.
if(x >= (-1) && x <= 1) {
for(k=0; sum!=sum_old; k++) {
sum_old = sum;
sum += pw((-1), k) * pw(x, (2*k) + 1)/((2*k) + 1);
count++;
printf("%d || %.17lf\n", count, sum);
}
Comparing doubles can be tricky. The conventional way to compare doubles is to test within epsilon. There should be an epsilon value defined somewhere, but for your purposes how many digits are enough to approximate? If you only need like 3 or 4 digits you can instead have
#define EPSILON 0.0001 //make this however precise you need to approximate.
if(x >= (-1) && x <= 1) {
for(k=0; fabs(sum - sum_old) > EPSILON; k++) {
sum_old = sum;
sum += pw((-1), k) * pw(x, (2*k) + 1)/((2*k) + 1);
count++;
printf("%d || %.17lf\n", count, sum);
}
If the issue is that -1,1 iterate too many times either reduce the precision or increase the step per iteration. I am not sure that is what you're asking though, please clarify.
I think the cause of this is for a mathematical reason rather than a programming one.
Away from the little mistakes and adjustments that you should do to your code, putting x = 1 in the infinite series of arctan, is a boundary condition:
In this series, we add a negative value to a positive value then a negative value. This means the sum will be increasing, decreasing, increasing, ... and this will make some difference each iteration. This difference will be smaller until the preciseness of double won't catch it, so the program will stop and give us the value.
But in the sum equation. When we set z = 1 and n goes from 0 to ∞, this will make this term (-1^n) equal to 1 in one time and -1 in the next iteration. Also,
the value of the z-term will be one and the denominator value when n approaches infinity will = ∞ .
So the sum several iterations will be like +1/∞ -1/∞ +1/∞ -1/∞ ... (where ∞ here represents a big number). That way the series will not reach a specific number. This is because z = 1 is a boundary in this equation. And that is causing infinite iterations in your solution without reaching a number.
If you need to calculate arctan(1) I think you should use this formula:
All formulas are from this Wikipedia article.
Here is some modifications that make your code more compact and has less errors:
#include <stdio.h>
#include <math.h>
#define x 0.5 //here x is much easier to change
double pw(double, double); //declaration of the function should be done
int main() { //the default return type of main is int.
double k;
double sum = 0 ; //you should initiate your variables.
double sum_old = 1 ; //=1 only to pass the for condition first time.
//you don't need to define counter here
if(x < -1 || x > 1){
printf("x is not defined. Please choose an x in the interval [-1, 1]\n");
return 0;
}
for(k=0; sum!=sum_old; k++) {
sum_old = sum;
sum += pw((-1), k) * pw(x, (2*k) + 1)/((2*k) + 1);
printf("%.0f || %.17lf\n", k, sum);
}
printf("My result is: %.17lf\n",sum);
printf("atan(%f) is: %.17f\n", x, atan(x));
printf("My result minus atan(x) = %.17lf\n", sum - atan(x));
return 0;
}
double pw(double y, double n) { //functions should be declared out of the main function
double i;
double number = 1;
for (i = 0; i < n; i++) {
number *= y;
}
return(number);
}
double fc (double y) {
double i;
double number = 1;
for (i = 1; i <= y; i++){
number *= i;
}
return(number);
}

-nan return value / e (euler) raised to a power calculation loop

I'm learning C programming and made the algorithm below to solve this problem:
The code actually works, but initially the loop was with only 10 repetitions (rep <= 10), and the anwer for p = 3 was almost correct, so I changed rep <= 20. And It gave me just the exact answer from my calculator. And then I tried with a higher number, 12, and the output again was inaccurate. So I ended raising rep <= 35. If I get the loop for higher repetitions I get "-nan", and if the input for p is too high it will be the same. So just have to see the pattern to know that the problem of inaccuracy will get back as I input higher numbers which is not the case because the output will be NaN if I input a high value.
Is it possible to solve it without higher level functions? just want to know if my program is ok for the level in which I am now...
#include <stdio.h>
int main()
{
float p; //the power for e
float power; //the copy of p for the loop
float e = 1; //the e number I wanna raise to the power of p
int x = 1; //the starting number for each factorial generation
float factorial = 1;
int rep = 1; //the repeater for the loop
printf( "Enter the power you want to raise: " );
scanf( "%f", &p );
power = p;
while ( rep <= 35) {
while ( x > 1) {
factorial *= x;
x--;
}
e += p / factorial;
//printf("\nthe value of p: %f", p); (TESTER)
//printf("\nthe value of factorial: %f", factorial); (TESTER)
p *= power; //the new value for p
rep++;
factorial = 1;
x = rep; //the new value for the next factorial to be generated
//printf("\n%f", e); (TESTER)
}
printf("%.3f", e);
return 0;
}
Sorry if I had syntax/orthography errors, I'm still learning the language.
Before we begin, let's write your original code as a function, with some clean-ups:
float exp_original(float x, int rep = 35)
{
float sum = 1.0f;
float power = 1.0f;
for (int i = 1; i <= rep; i++)
{
float factorial = 1.0f;
for (int j = 2; j <= i; j++)
factorial *= j;
power *= x;
sum += power / factorial;
}
return sum;
}
There were some unnecessary variables you used which were removed, but otherwise the procedure is the same: compute the factorial from scratch.
Let's look at the ratio between successive terms in the series:
We can thus simply multiply the current term by this expression to get the next term:
float exp_iterative(float x, int rep = 35)
{
float sum = 1.0f;
float term = 1.0f;
for (int i = 1; i <= rep; i++)
{
term *= x / i;
sum += term;
}
return sum;
}
Seems much simpler, but is it better? Comparison against the C-library exp function (which we assume to be maximally precise):
x exp (C) exp_orig exp_iter
-------------------------------------------
1 2.7182817 2.718282 2.718282
2 7.3890562 7.3890567 7.3890567
3 20.085537 20.085539 20.085539
4 54.598148 54.598152 54.598152
5 148.41316 148.41318 148.41316
6 403.4288 403.42871 403.42877
7 1096.6332 1096.6334 1096.6334
8 2980.958 2980.9583 2980.9587
9 8103.084 8103.083 8103.083
10 22026.465 22026.467 22026.465
11 59874.141 59874.148 59874.152
12 162754.8 162754.77 162754.78
13 442413.41 -nan(ind) 442413.38
14 1202604.3 -nan(ind) 1202603.5
15 3269017.3 -nan(ind) 3269007.3
16 8886111 -nan(ind) 8886009
17 24154952 -nan(ind) 24153986
18 65659968 -nan(ind) 65652048
19 1.784823e+08 -nan(ind) 1.7842389e+08
20 4.8516518e+08 -nan(ind) 4.8477536e+08
The two custom implementations are neck-and-neck in-terms of precision, until x = 13 where the original gives NaN. This is because the highest power term 13^35 = 9.7278604e+38 exceeds the maximum value FLT_MAX = 3.40282e+38. The accumulated term in the iterative version never reaches anywhere near the limit.

Calculate Factorial within a single "for" loop to calculate sum of series

It took me a while conceptual to grasp how to code a loop that would calculate a given series in which a factorial was used.
I coded it--then my teacher told us we had to use a single for loop. I can't seem to grasp how to do something like this. It doesn't make sense how you'd keep the running total of the products across several numbers.
Here is my code; which includes a nested for loop. I really appreciate any and all help.
int main() {
/*init variables*/
int N; //number of terms
float NUMER, DENOM = 1;
float FRAC, sum = 0, x;
/*asks user for value of N*/
printf("Input number of terms: ");
scanf("%i", &N);
/*asks user for value of x*/
printf("Input value of x: ");
scanf("%f", &x);
for (int n = 0; n <= N; n++) {
NUMER = (pow(x, n)); //calculates numerator
for (int fac = 1; fac <= n; fac++) { //calculates factorial using for loop
DENOM = n * fac;
}
if (DENOM <= 0)
printf("\n\nError, dividing by zero.\n\n"); //this is for debugging purposes; disregard
FRAC = NUMER / DENOM; //calculates fraction
sum += FRAC; //running sum of series
}
printf("\nSum of the series is %.1f\n\n", sum); //prints sum of series
return 0;
You want DENOM = n!, so you can just start with DENOM = 1
and update the value inside the loop:
DENOM = 1;
for (int n = 0; n <= N; n++) {
NUMER = (pow(x, n)); //calculates numerator
FRAC = NUMER / DENOM; //calculates fraction
sum += FRAC; //running sum of series
DENOM *= n+1;
}
Instead of computing x^n and n! each time through the outer loop, you can initialize
the quotient to 1.0 before the outer loop, then on each pass through the outer loop,
multiply by x/n to get the next term in the series. This will avoid the need
to call pow(x,n), and use an inner loop to calculate the factorial, each pass through
the outer loop.
If you think about what you would do if calculating a factorial by hand, I think you can figure out how to code this pretty easily.
Lets say you are trying to calculate 11!. Well, you would start at 11, and them multiply by 10. Now you have 110. Now multiply by 9. You have 990. Now multiply by 8...
As you can see, the 11, 10, 9, 8... series is what your for loop is going to be. Just keep your 'current answer' in a variable and keep multiplying it by the number provided by your for loop.
That seems...complicated. Terseness is or can be your friend :D
I don't think it needs to be much more complicated than:
#include <string.h>
#include <stdio.h>
#include <stdlib.h>
int main( int argc, char* argv[] )
{
double limit = 10 ; // how far do we want to go?
double x = 2 ; // some value for X
double xn = 1 ; // by definition, for all X, X^0 is 1
double nf = 1 ; // by convention, 0! is 1
double value = 0 ;
double sum = 0 ;
double n = 0 ;
while ( n < limit )
{
value = xn / nf ; // compute the next element of the series
sum += value ; // add that to the accumulator
xn *= x ; // compute the *next* value for X^n
nf *= (++n) ; // compute the *next* value for N!
}
return 0;
}
You get a more stable answer working the loop in reverse. Many infinite sums numerically come out better summing the smallest terms together first.
f(x,n) = x^0/0! + x^1/1! + x^2/2! + ... + x^n/n!
Let the sum be S(x,n) = x/n
Let the sum of the 2 last terms be S(x,n-1) = x/(n-1) + x/(n-1)*S(x,n)
Let the sum of the 3 last terms be S(x,n-2) = x/(n-2) + x/(n-2)*S(x,n-1)
...
Let the sum of the N last terms be S(x,1) = x/(1) + x/(1)*S(x,1)
double e(double x, unsigned n) {
double sum = 0.0;
while (n > 0) {
sum = x*(1 + sum)/n;
n--;
}
sum += 1.0; // The zero term
return sum;
}
Notice that even if n is large like 1000, and the mathematical answer < DBL_MAX, this loop does not run into floating point overflow so easily.
[edit] But if code must be done in a forward loop, the below calculates each term not as separate products that may overflow, but a unified computation.
double e_forward(double x, unsigned n) {
double sum = 1.0;
double term = 1.0;
for (unsigned i = 1; i <= n; i++) {
term *= x / i;
sum += term;
}
return sum;
}

The outermost for loop does not work as intended

I have been using Ubuntu 12.04 LTS with GCC to compile my the codes for my assignment for a while. However, recently I have run into two issues as follows:
The following code calculates zero for a nonzero value with the second formula is used.
There is a large amount of error in the calculation of the integral of the standard normal distribution from 0 to 5 or larger standard deviations.
How can I remedy these issues? I am especially obsessed with the first one. Any help or suggestion is appreciated. thanks in advance.
The code is as follows:
#include <stdio.h>
#include <math.h>
#include <limits.h>
#include <stdlib.h>
#define N 599
long double
factorial(long 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;
//Here pro stands for product.
if (n < 0)
printf("Factorial is not defined for a negative number \n");
else {
while (n >= s) {
pro *= s;
s++;
}
return pro;
}
}
int main()
{
// Since the function given is the standard normal distribution
// probability density function we have mean = 0 and variance = 1.
// Hence we also have z = x; while dealing with only positive values of
// x and keeping in mind that the PDF is symmetric around the mean.
long double * summand1 = malloc(N * sizeof(long double));
long double * summand2 = malloc(N * sizeof(long double));
int p = 0, k, z[5] = {0, 3, 5, 10, 20};
long double sum1[5] = {0}, sum2[5] = {0} , factor = 1.0;
for (p = 0; p <= 4; p++)
{
for (k = 0; k <= N; k++)
{
summand1[k] = (1 / sqrtl(M_PI * 2) )* powl(-1, k) * powl(z[p], 2 * k + 1) / ( factorial(k) * (2 * k + 1) * powl(2, k));
sum1[p] += summand1[k];
}
//Wolfamalpha site gives the same value here
for (k = 0; k <= N; k++)
{
factor *= (2 * k + 1);
summand2[k] = ((1 / sqrtl(M_PI * 2) ) * powl(z[p], 2 * k + 1) / factor);
//printf("%Le \n", factor);
sum2[p] += summand2[k];
}
sum2[p] = sum2[p] * expl((-powl(z[p],2)) / 2);
}
for (p = 0; p < 4; p++)
{
printf("The sum obtained for z between %d - %d \
\nusing the first formula is %Lf \n", z[p], z[p+1], sum1[p+1]);
printf("The sum obtained for z between %d - %d \
\nusing the second formula is %Lf \n", z[p], z[p+1], sum2[p+1]);
}
return 0;
}
The working code without the outermost for loop is
#include <stdio.h>
#include <math.h>
#include <limits.h>
#include <stdlib.h>
#define N 1200
long double
factorial(long 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;
//Here pro stands for product.
if (n < 0)
printf("Factorial is not defined for a negative number \n");
else {
while (n >= s) {
pro *= s;
s++;
}
return pro;
}
}
int main()
{
// Since the function given is the standard normal distribution
// probability density function we have mean = 0 and variance = 1.
// Hence we also have z = x; while dealing with only positive values of
// x and keeping in mind that the PDF is symmetric around the mean.
long double * summand1 = malloc(N * sizeof(long double));
long double * summand2 = malloc(N * sizeof(long double));
int k, z = 3;
long double sum1 = 0, sum2 = 0, pro = 1.0;
for (k = 0; k <= N; k++)
{
summand1[k] = (1 / sqrtl(M_PI * 2) )* powl(-1, k) * powl(z, 2 * k + 1) / ( factorial(k) * (2 * k + 1) * powl(2, k));
sum1 += summand1[k];
}
//Wolfamalpha site gives the same value here
printf("The sum obtained for z between 0-3 using the first formula is %Lf \n", sum1);
for (k = 0; k <= N; k++)
{
pro *= (2 * k + 1);
summand2[k] = ((1 / sqrtl(M_PI * 2) * powl(z, 2 * k + 1) / pro));
//printf("%Le \n", pro);
sum2 += summand2[k];
}
sum2 = sum2 * expl((-powl(z,2)) / 2);
printf("The sum obtained for z between 0-3 using the second formula is %Lf \n", sum2);
return 0;
}
I'm quite certain that the problem is in factor not being set back to 1 in the outer loop..
factor *= (2 * k + 1); (in the loop that calculates sum2.)
In the second version provided the one that works it starts with z=3
However in the first loop since you do not clear it between iterations on p by the time you reach z[2] it already is a huge number.
EDIT: Possible help with precision..
Basically you have a huge number powl(z[p], 2 * k + 1) divided by another huge number factor. huge floating point numbers lose their precision. The way to avoid that is to perform the division as soon as possible..
Instead of first calculating powl(z[p], 2 * k + 1) and dividing by factor :
- (z[p]z[p] ... . * z[p]) / (1*3*5*...(2*k+1))`
rearrange the calculation: (z[p]/1) * (z[p]^2/3) * (z[p]^2/5) ... (z[p]^2/(2*k+1))
You can do this in sumand2 calculation and a similar trick in summand1

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