📘 numerical methods

1. **State the problem:** We want to compute the second approximation $y_2$ for the initial value problem $$y' = e^{-t} + 5y^2$$ with initial condition $y(0) = 0.00$, step size $h=

1. **State the problem:** We want to find an approximation to the cube root of 7, i.e., find $x$ such that $x^3 = 7$. 2. **Define the function:** Let $f(x) = x^3 - 7$. We want to f

1. **State the problem:** We want to approximate $\sqrt{7}$ using the Newton-Raphson method starting with $x_0 = 1.3$ and find the value correct to six decimal places. 2. **Define

1. **State the problem:** We want to find the 5th root of a number $n$ using the Newton-Raphson method, starting with an initial guess $x_0=1.5$. Specifically, estimate the 5th roo

1. **State the problem:** We need to find the approximate area of the cross section of a river using Simpson's one-third rule given the depth $y$ at distances $x$ from one bank. 2.

1. The problem is to solve a system of linear equations using the numerical method BCA (Backward, Central, and Adams methods are common numerical methods, but BCA is not standard;

1. **Problem Statement:** Find the real root of the polynomial function $$f(x) = -26 + 85x - 91x^2 + 44x^3 - 8x^4 + x^5$$ using three methods:

1. **Problem Statement:** Obtain the LU factorization of the matrix $$A = \begin{pmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \end{pmatrix}$$

1. The user asks for help with numerical method questions but does not specify any particular problem. 2. Since no specific numerical method problem is provided, I cannot solve or

1. **Problem 1: Solve $f(x) = x^4 - 4x - 9 = 0$ using Newton-Raphson and Secant Methods with given initial guesses.** 2. **Newton-Raphson Method:**

1. **Problem:** Find the approximate root at the 3rd iteration for the equation $f(x) = 2x^3 + 5x + 20 = 0$ using the secant method, then determine the relative error. 2. **Formula

1. The problem asks when Newton's Method fails to find an approximate solution of an equation. 2. Newton's Method uses the formula:

1. **State the problem:** We want to find an estimated root of the nonlinear equation $$e^{\cos(x+2)} = \ln(2 - x^2) + 2$$ near the initial guess $$x_0 = -0.7$$ using Newton's meth

1. **Problem Statement:** Find the value of $y$ at $x=0.1$ using Euler's modified method with step size $h=0.1$ for the differential equation $$\frac{dy}{dx} = 1 - y, \quad y(0) =

1. **Problem Statement:** Jelaskan apa yang dimaksud dengan metode eliminasi dan metode iterasi pada masalah sistem persamaan numerik. 2. **Metode Eliminasi:** Metode eliminasi ada

1. Masalah: Jelaskan apa yang dimaksud dengan metode eliminasi dan metode iterasi pada sistem persamaan numerik. 2. Metode Eliminasi:

1. **Problem Statement:** We are given a system of linear equations (SPL): $$\begin{cases} 3x - y + z = 5 \\ x + 4y + 2z = 6 \\ 2x - y + 5z = 7 \end{cases}$$

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