ЛБ6 / Звіт

Purpose of the work: application of numerical methods of calculating derivatives in various practical problems. Analysis and comparison of results with analytical ones.

Work progress

1.Choose an option of an individual task from the table. 4.1 according to the number n of the student in the group list.

Variant 11: ( ) = − − + − −

2.Using the rules of analytical calculation of derivatives, calculate the derivative f`(x) and get an exact solution to the problem.

`( ) = (−4 4 − 3 + 2 − 4 − 4)` = − − + −

b. Calculate function and derivative values at nodes .

4.Develop an algorithm for the numerical differentiation method using the simplest numerical differentiation formulas.

Impement the function to find first derivative, using backward difference, central difference and second backward difference formulas.

• Backward difference: `( ) = ( + )− ( )

static double backward_difference(double x)

{

return (f(x + h) - f(x)) / h;

}

Where ‘h’ is the constant value set by developer manually.

• Central difference: `( ) = ( + )− ( − )

2

static double central_difference(double x)

{

return (f(x + h) - f(x - h)) / 2. * h;

}

• Second backward difference: ``( ) = ( + )−2 ( )+( − )

2

static double second_central_difference(double x)

{

return (f(x + h) – 2. * f(x) + f(x - h)) / pow(h, 2);

}

5.For the constructed nodes, using the developed program, find the value of the first and second derivatives according to formulas (4.1) - (4.3). In each case, estimate the differentiation error in two ways: 1) as the modulus of the

difference between the exact and approximate values of the derivative; 2) using the appropriate formulas for derivative errors:

a.First we must implement the functions to find the absolute and marginal errors of each method.

•Absolute error calculation function.

//The coefficients of my polynomial constexpr double a = -4;

constexpr double b = -1; constexpr double c = 1; constexpr double d = -4; constexpr double e = -4;

//General formula to find the derivative of fourth //degree polynomial static double first_derivative(double x)

{

return 4 * a * pow(x, 3) + 3 * b * pow(x, 2) + 2 * c * x + d;

}

template<typename Method, typename Derivative>

static double absolute_error_calculation(Method formula, Derivative der, double x)

{

return std::abs(formula(x) - der(x));

}

• Now define functions to find marginal errors of each method.

Formula to find marginal error for difference derivative:

The coding implementation worth a remark. To find maximum value of fourth degree polynomials second derivative, it requires finding only one critical point of the one that is actually calculated by solving an equation 24 + 6 = 0 and the critical

point can be found as − 246 = − 4 , check if critical point is at the interval and then

determine the maximum second derivative’s absolute value at left, right limits and the critical point.

static double calculate_M1(double x0, double x1)

{

double crit_point = -b / (4. * a);

if (crit_point < x0 || crit_point > x1) crit_point = x0;

d. Program listing.

#include <iostream>

constexpr double a = -4; constexpr double b = -1; constexpr double c = 1; constexpr double d = -4; constexpr double e = -4;

constexpr double h = 1;

static double f(double x)

{

return a * pow(x, 4) + b * pow(x, 3) + c * pow(x, 2) + d * x + e;

}

static double first_derivative(double x)

{

return 4 * a * pow(x, 3) + 3 * b * pow(x, 2) + 2 * c * x + d;

}

static double second_derivative(double x)

{

return 12 * a * pow(x, 2) + 6 * b * x + 2 * c;

}

static double third_derivative(double x)

{

return 24 * a * x + 6 * b;

}

static double fourth_derivative(double x)

{

return 24 * a;

}

static double difference_derivative(double x)

{

return (f(x + h) - f(x)) / h;

}

static double central_difference_derivative(double x)

{

return (f(x + h) - f(x - h)) / (2. * h);

}

static double second_difference_derivative(double x)

{

return (f(x - h) - 2 * f(x) + f(x + h)) / pow(h, 2);

}

template<typename Method, typename Derivative>

static double absolute_error_calculation(Method formula, Derivative der, double x)

{

return std::abs(formula(x) - der(x));

}

static double calculate_M1(double x0, double x1)

{

double crit_point = -b / (4. * a);

if (crit_point < x0 || crit_point > x1) crit_point = x0;

return std::max({ std::abs(second_derivative(x0)), std::abs(second_derivative(x1)), std::abs(second_derivative(crit_point)) });

}

static double marginal_error_for_difference_derivative(double x)

{

double x0 = x; double x1 = x + h;

return calculate_M1(x0, x1) * h / 2.;

}

static double marginal_error_for_central_difference(double x)

{

double x0 = x - h; double x1 = x + h;

double M2 = std::max(std::abs(third_derivative(x0)), std::abs(third_derivative(x1)));

return M2 * pow(h, 2) / 6.;

}

static double marginal_error_for_second_difference(double x)

{

return pow(h, 2) / 12. * std::abs(fourth_derivative(24 * a)) + 0.000000001;

}

static void calculate_values()

{

double exact_value, value, absolute_error, marginal_error, x, x0 = 0; int i;

static auto print = [&](std::string str) { std::cout << "x_" << i << " = " << x <<

"\n" << str <<

"\n\tExact value: " << exact_value << "\n\tValue: " << value <<

"\n\tAbsolute error: " << absolute_error << "\n\tMarginal error: " << marginal_error <<

"\nDoes absolute error exceed marginal error? - " << std::boolalpha << (absolute_error > marginal_error) << "\n\n"; };

for (i = -2; i <= 2; ++i)

{

x = x0 + i * h;

exact_value = first_derivative(x); value = difference_derivative(x);

absolute_error = absolute_error_calculation(difference_derivative, first_derivative, x);

marginal_error = marginal_error_for_difference_derivative(x); print("difference derivative");

exact_value = first_derivative(x);

value = central_difference_derivative(x);

absolute_error = absolute_error_calculation(central_difference_derivative, first_derivative, x);

marginal_error = marginal_error_for_central_difference(x); print("central difference derivative");

exact_value = second_derivative(x); value = second_difference_derivative(x);

absolute_error = absolute_error_calculation(second_difference_derivative, second_derivative, x);

marginal_error = marginal_error_for_second_difference(x); print("second difference derivative");

}

}

int main()

{

calculate_values();

return 0;

}

b.Change the h value in our program just altering the const value ‘h’ to 0.1

c.Find values and errors.