Lab 3. Solution Methods of Non-Linear Equations.
Write a program to
determine the real root of the function
.
The program should also compute and output the true error
after each iteration. The stopping criterion is
.
Variant
1. Find
the real root of
using
the bisection method.
Variant
2. Find
the real root of
using the secant method.
Variant
3. Find
the real root of
using the false
position method.
Variant
4. Find
the real root of
using the Newton-Raphson
method.
Lab 4. Integration of Ordinary Differential Equations.
Write a user-friendly computer program to implement any numerical methods for solving a single ordinary differential equation (ODE). Check your results using the Runge's formula:
.
In
this formula values
at
are computing with a step size
and
correspondingly;
(Euler
and midpoint method),
(classic
fourth – order RK and fourth – order Adams methods).
Variant
1. Solve
the following problem numerically from
;
.
Use the Euler's method with a step size of 0.5.
Variant
2. Solve
the following problem numerically from
;
.
Use the midpoint method with a step size of 0.1.
Variant
3. Solve
the following problem numerically from
;
.
Use the fourth – order RK method with a step size of 0.2.
Variant
4. Solve
the following problem numerically from
;
.
Use the fourth – order Adams method. Employ a step size of 0.05 and
the Euler's method to predict the start-up values.
Variant
5. Solve
the following problem numerically from
;
.
Use the fourth – order Adams method. Employ a step size of 0.1 and
the fourth – order RK method to predict the start-up values.
Lab 5. Interpolation and Extrapolation of Functions.
Develop, debug, and test a user-friendly computer program to implement any numerical methods to estimate intermediate values between precise data points:
x |
0.48 |
0.55 |
0.60 |
0.72 |
0.90 |
1.20 |
1.67 |
1.95 |
2.20 |
2.22 |
2.50 |
f(x) |
1.616 |
1.733 |
1.822 |
2.054 |
2.459 |
3.320 |
5.312 |
7.028 |
9.025 |
9.207 |
12.182 |
Variant
1. Calculate
using Lagrange interpolation polynomial.
Variant 2. Calculate using the Newton's divided difference interpolation polynomial.
Variant
3. Determine the value of x
that corresponds to
.
Employ inverse interpolation using Newton (Gregory-Newton)
interpolation polynomial.
Variant
4. Develop a) linear splines and b) quadratic
splines for the 11 data points, predict
and
.
Variant 5. Develop cubic splines for the 11 data points, predict and .
Julius J. Katsman
Numerical Methods
Workbook
Reviewed by: V.G. Spitsyn, Professor of the Automation and Computer Technology Department, TPU, D.Sc.
Editor: A.Y. Tsyba