Numerical Methods Labs

Four labs of a numerical methods course: series approximation, root finding, linear systems and numerical integration, each solved for several task variants in C++, Python, C and C# (file names use AMO, the course abbreviation).

Year: 2021 · Language: C++ (also Python, C, C#) · Context: university coursework (KPI)

What's inside

Each lab has one folder per variant (varN, varN_k = an alternative solution of the same variant). Task texts are not in the repo; the descriptions below are read from the code.

Lab	What it does	Variants (folders)
lab1/	Taylor-series evaluation of ln x, ch x, sh x, sin x with stopping by epsilon (1e-2 ... 1e-14); prints a table of n, absolute error and remainder term, then a second table over 11 points of the interval for the n found at eps = 1e-8	var1, var7, var12, var14, var14_1, var15 (C#), var17, var21
lab2/	Root of a nonlinear equation: simple iteration, Newton, chord methods; error estimate and iteration count per epsilon	var1, var3_0 (Py), var3_1, var6, var7, var12, var15_0, var15_1/var15_2 (C#), var21, var27 (C)
lab3/	4x4 linear system: Gauss-Jordan elimination vs. iterative methods (direct iteration, Seidel)	var4, var7, var7_1, var12, var13, var20
lab4/	Definite integrals: Simpson and trapezoid rules, step chosen from the error bound, comparison with the Newton-Leibniz value, Runge error estimate	var3, var5, var7, var8, var12, var15, var20, var21, var26

Interesting bits

• lab1/var1/var1.cpp computes ln x with range reduction: finds m with 2^m >= x, sets z = x / 2^m and sums the series in (1 - z) / (1 + z), then reassembles m*ln2 - 2*sum. lab1/var15/TaskSolver.cs does the same kind of reduction for sin by subtracting multiples of pi and tracking the sign.
• lab2/var3_0/var3.py implements simple iteration with lambda = 1/M1 and contraction factor q = 1 - m1/M1, using the a-posteriori stop q*|xn - x0| / (1 - q) < eps; lab2/var6/Chord.cpp picks the fixed end by the sign of f(a)*f''(a) and stops on |f(xn)| / m1 < eps.
• lab2/var27/var27.c separates roots first: it scans the interval with step 0.01, collects sign-change intervals where the derivative keeps its sign into a hand-written linked list, and refines each with bisection or iteration.
• lab4/var12/main.cpp derives the trapezoid step directly from the error bound (h = sqrt(12*eps / ((b-a)*M))); lab4/var21/Task.cpp has a runge() function that doubles n until the Runge estimate |I(n) - I(2n)| / (2^r - 1) drops below the target.

Run

Verified (build and run) with g++ 13.3 on Linux, one variant folder at a time:

g++ -std=c++14 lab1/var12/*.cpp -o var12 && ./var12     # lab1/var1, var7, var12, var14*, var17, var21; lab2/var1, var6, var7, var12, var15_0, var21, var3_1; lab3/var12
g++ -std=c++17 -fpermissive -include cmath lab4/var12/*.cpp -o var12 && ./var12    # lab3/var7, var7_1, lab4/*
gcc lab2/var27/var27.c -lm -o var27 && ./var27
python3 lab4/var3/main.py                                # all .py files run; lab3/var4 asks for epsilon on stdin

The -include cmath flag is needed because several files rely on a Visual Studio header that pulls in <cmath> implicitly; lab3/var7* use std::size (C++17). The C# files (Program.cs, Task*.cs, TaskSolver.cs) were not compiled (no .NET toolchain here).

Note

Written in 2021, before AI coding assistants. Kept as-is for history.