📘 numerical analysis

1. The problem is to solve a system of differential equations numerically, often written as $$\frac{d\mathbf{y}}{dt} = \mathbf{f}(t, \mathbf{y})$$ where $\mathbf{y}$ is a vector of

1. **Problem Statement:** Solve the differential equation $$y'' + y = 2e^x$$ and compute $$y(0.4)$$ using Milne's method given the initial table:

1. **Problem Statement:** Find all eigenvalues and corresponding eigenvectors of the matrix $$A = \begin{bmatrix}4 & 3 & 2 & 1 \\ 3 & 4 & 3 & 2 \\ 2 & 3 & 4 & 3 \\ 1 & 2 & 3 & 4\en

1. **State the problem:** We need to find the relative error when approximating the expression $5x^2 - 1$ using 4-digit chopping for $x = 11110.3338$. 2. **Recall the formula for r

1. **State the problem:** We want to find the relative error when approximating the expression $5x^2 - 1$ using 4-digit chopping for $x = 11110.3338$. 2. **Recall the formula for r

1. **State the problem:** We want to find the relative error when approximating the expression $5x^2 - 1$ using 4-digit chopping, where $x = 11110.3338$. 2. **Recall the formula fo

1. **Problem Statement:** (a) Estimate $f(0.125)$ using Newton's forward interpolation formula given the data:

1. **Problem Statement:** (a) Construct a forward difference table for given $x$ and $f(x)$ values.

1. **Problem 1:** Find the roots of the quadratic equation $$\frac{1}{3}x^2 + \frac{123}{4}x - \frac{1}{6} = 0$$ using four-digit rounding arithmetic and the quadratic formula. The

1. **State the problem:** We want to solve the nonlinear system using Newton's method: $$\begin{cases} 10x_1 - 2x_2^2 + x_2 - 2x_3 - 5 = 0 \\ 8x_2^2 + 4x_3^2 - 9 = 0 \\ 8x_2 x_3 +

1. The problem is to verify if the given table of nodes $x_k$ and weights $\lambda_k$ corresponds correctly to the Gauss-Chebyshev quadrature method for different values of $n$. 2.

1. **Énoncé du problème** : Trouver la fonction $u(x,y)$ solution du problème de Dirichlet sur le domaine $\Omega = (0,1) \times (0,1)$ : $$-\Delta u(x,y) = f(x,y), \quad (x,y) \in

1. **Problem Statement:** We want to prove the error estimate lemma for B-spline collocation:

1. **Statement of the problem:** We want to prove the error estimate lemma for B-spline collocation, which states that for a function $y(t) \in C^{p+1}[a,b]$ and an approximate sol

1. **State the problem:** We are given data points $(x, f(x))$ as $(25,5)$, $(36,6)$, $(49,7)$, $(64,8)$, and $(81,9)$. We want to construct a natural cubic spline that fits these

1. **Fit a straight line $y = a + bx$ using least squares for the data:** $x$: 2, 4, 6, 8, 10

1. **Evaluate** $\int_0^6 (x^2 + 3x) \, dx$ using Simpson's 1/3 rule with 6 equal parts. - Interval $[0,6]$, divide into 6 parts: $h = \frac{6-0}{6} = 1$

1. We are given two sets of data points and two models for approximation by the least-squares method: - Model 1: $$\varphi(x) = c_1 \frac{1}{x} + c_2$$ with points $$(-3, 32.3), (-

1. Построим интерполяционный многочлен Ньютона с раздельными разностями для данных точек $x = 0,1,2,3,4$ и значений функции $f(0)=2, f(1)=0, f(2)=1, f(3)=2, f(4)=2$.\n\n2. Сначала

1. The problem involves applying Newton's Method (NM) to find the root of the function defined by its derivative $$f'(x) = x - e^{-x}$$ and second derivative $$f''(x) = 1 + e^{-x}$

1. We are given the problem to find the root of the function $f(x) = x^2 - 3$ in the interval $[1,2]$ using the bisection method with an error tolerance $\epsilon = 0.01$. 2. The b