[2022 Fall] Assignment2-7¶

Course: AP3021

Question: $x = \dfrac{x + \frac{a}{x}}{2}$, then find the root of a.¶

Answer:¶

In [1]:

# First through the machine eplison. (HW2-5)

def machineEpsilon() :
    epsilon = 1.0

    while (epsilon > 0) :
        xmin = epsilon
        epsilon = epsilon / 2

    return xmin


def IterMeth(val, es, maxIterator) :
    iter = 1
    sol = val
    ea = 100
    count = 1

    while True :
        preSol = sol
        sol = (sol + (val / sol)) / 2
        iter = iter + 1

        if sol != 0 :
            ea = abs((sol - preSol) / sol) * 100      

        if (ea < es or iter >= maxIterator) :
            break

        print(f"Times: {count}, Root: {sol :.4f}; Approximate relative error: {ea :.4f}%.")

        count += 1

    print("\nThe evaluate root:", sol)

    return sol

print("Machine eplison:", machineEpsilon())
print("What the value of \"a\" we picked up -> 25\n")
print("Start evaluate: ")

IterMeth(25, machineEpsilon(), 100)

# First through the machine eplison. (HW2-5) def machineEpsilon() : epsilon = 1.0 while (epsilon > 0) : xmin = epsilon epsilon = epsilon / 2 return xmin def IterMeth(val, es, maxIterator) : iter = 1 sol = val ea = 100 count = 1 while True : preSol = sol sol = (sol + (val / sol)) / 2 iter = iter + 1 if sol != 0 : ea = abs((sol - preSol) / sol) * 100 if (ea < es or iter >= maxIterator) : break print(f"Times: {count}, Root: {sol :.4f}; Approximate relative error: {ea :.4f}%.") count += 1 print("\nThe evaluate root:", sol) return sol print("Machine eplison:", machineEpsilon()) print("What the value of \"a\" we picked up -> 25\n") print("Start evaluate: ") IterMeth(25, machineEpsilon(), 100)

Machine eplison: 5e-324
What the value of "a" we picked up -> 25

Start evaluate: 
Times: 1, Root: 13.0000; Approximate relative error: 92.3077%.
Times: 2, Root: 7.4615; Approximate relative error: 74.2268%.
Times: 3, Root: 5.4060; Approximate relative error: 38.0226%.
Times: 4, Root: 5.0152; Approximate relative error: 7.7918%.
Times: 5, Root: 5.0000; Approximate relative error: 0.3045%.
Times: 6, Root: 5.0000; Approximate relative error: 0.0005%.
Times: 7, Root: 5.0000; Approximate relative error: 0.0000%.

The evaluate root: 5.0

Out[1]:

5.0

C Language Version¶

#include <stdio.h>
#include <math.h>

double preAbs(double), machineEpsilon(void), IterMeth(double, double, double);

double preAbs(double x) {
  if (x < 0)
    x = -x;

  return x;
}

double machineEpsilon(void) {
  double epsilon = 1.0, xmin;

  while (epsilon > 0) {
    xmin = epsilon;
    epsilon = epsilon / 2;
  }

  return xmin;
}

double IterMeth(double val, double es, double maxit) {
  double iter = 0.0, sol = val, ea = 100.0, solold;

  while (1) {
    solold = sol;
    sol = (sol + val / sol) / 2;
    iter += 1.0;

    if (sol != 0)
      ea = preAbs((sol - solold) / sol) * 100;

    printf("%g %g\n", sol, ea);

    if ((ea <= es) || (iter >= maxit))
      break;
  }

  return sol;
}

int main() {
  double a;

  printf("Input a: ");
  scanf("%lf", &a);
  printf("%g", IterMeth(a, machineEpsilon(), 100));
  printf("\n");

  return 0;
}

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Last update: 2024-04-27