📘 numerical methods

1. **Problem Statement:** Solve the system of linear equations using the Relaxation Method starting with the initial vector $(0,0,0)$.

1. **Problem Statement:** We need to find a root of the equation $$x^3 + x^2 + x + 7 = 0$$ using the Secant Method with four iterations. 2. **Definition of Secant Method:** The Sec

1. **State the problem:** We want to find the approximated root after the second iteration of the Newton-Raphson method for the equation $$x^3 + 4x^2 - 10 = 0$$ starting with the i

1. The problem asks for the primary condition to apply the bisection method to find a root of a function $f(x)$. 2. The bisection method requires that the function $f(x)$ is contin

1. The problem asks to identify a disadvantage of the Secant Method compared to Newton's Method. 2. Recall the key characteristics:

1. **Problem 1:** Solve the equation $$f(x) = x^3 + x^2 - 3x - 3 = 0$$ using the False Position Method to find a positive real root. 2. **False Position Method Formula:**

1. Masalah yang diberikan adalah menyelesaikan persamaan diferensial menggunakan metode Runge-Kutta orde 4. 2. Metode Runge-Kutta orde 4 digunakan untuk menghitung solusi numerik d

1. Masalah yang diberikan adalah persamaan rekursif untuk $u^{n+1}$ yang melibatkan beberapa variabel dan operator matriks. 2. Persamaan yang diberikan adalah:

1. El problema es encontrar la raíz de una función usando el método de Newton-Raphson. 2. La fórmula del método de Newton-Raphson es:

1. **Problem Statement:** Find the root of the equation $$x^3 - 2x - 5 = 0$$ using the bisection method correct up to 3 decimal places. 2. **Formula and Method:** The bisection met

1. Задачата е да намерим корените на уравнението $$x^3 - 3x^2 + 3 = 0$$ с точност $$e = 0.00001$$, използвайки метода на секущите в интервала $$[1, 2]$$. 2. Методът на секущите изп

1. **Problem Statement:** Find the root of the function $f(x) = e^x - 2x^2$ in the interval (1, 2) using the Newton-Raphson method up to 6 iterations and calculate the correspondin

1. **Problem:** Use the bisection method to find a root of the equation $x^3 - 4x - 9 = 0$ in the interval $(2, 3)$ up to 5 iterations, and calculate the corresponding error for ea

1. **Problem statement:** Use the 4th order Runge-Kutta method to solve the differential equation $$\frac{dy}{dx} = y - \frac{x}{y}$$ with initial condition $$y(0) = 0$$ at $$x = 0

1. **Problem Statement:** Explain the methods to find roots of equations and solve two example problems using these methods. 2. **Mathematical Background:** Root-finding methods ai

1. **Problem statement:** We want to find a root of the function $f(x) = x^3 - 4x^2 - 10$ in the interval $[1,2]$ using fixed point iteration. 2. **Fixed point iteration method:**

1. **Problem statement:** Estimate $f(2.15)$ and $f(2.9)$ using Newton's finite difference interpolation based on the given data points: $$\begin{array}{c|c}

1. **Problem Statement:** Estimate the values of $f(2.15)$ and $f(2.9)$ using Newton's finite difference interpolation formulas given a table of values. 2. **Formula and Explanatio

1. **Problem statement:** Estimate $f(2.15)$ and $f(2.9)$ using the given data and Newton's finite difference interpolation formulas. 2. **Given data:**

1. **Problem a:** Calculate the integral $$\int_0^1 \frac{1}{1+x^2} dx$$ using the Trapezoidal rule with 6 intervals. 2. **Formula:** The Trapezoidal rule for $n$ intervals is give

1. **Problem Statement:** Use Newton's iteration method to find the root of a function $f(x)$ starting from an initial guess $x_0$. 2. **Formula:** Newton's iteration formula is gi