📘 numerical methods

1. **State the problem:** We need to find roots of the equations using the Regula-Falsi Method correct to four decimal places. 2. **Equations:**

1. **State the problem:** Use the Bisection Method to find a root of the equation $$3x^3 + 5x - 40 = 0$$ correct to three decimal places. 2. **Formula and method:** The Bisection M

1. Statement of the problem: Use the Newton-Raphson method to find a root of the function $f(x)=x^3-5x^2-4x$ near the initial guess $x_0=5$.\n2. Formula and derivative: The Newton-

1. **Problem 1: Find the approximate root of $f(x) = x^3 + 2x^2 - 1$ between 0 and 1 using the bisection method after four iterations.** The bisection method formula is: $$C = \fra

1. **Problem Statement:** Use the bisection method to find an approximate root of the function $$f(x) = x^3 - x - 2$$ in the interval $$[1, 2]$$ with 4 iterations. 2. **Formula and

1. **Problem 1: Factor matrix $A$ into $LL^T$ form with $L_{11} = 1$, where $L$ is lower triangular with ones on the diagonal, and $U$ is upper triangular.** Given:

1. **Convert decimal 0.6402 to binary (5 steps):** To convert a decimal fraction to binary, multiply by 2 repeatedly and record the integer part each time.

1. **State the problem:** We are given the function $f(x) = x^3 - 3x + 1$ and initial points $x_0 = 0$ and $x_1 = 1$. We need to use the method of False Position (Regula Falsi) to

1. The bisection method is a numerical technique to find roots of a continuous function $f(x)$ where the function changes sign over an interval $[a,b]$. 2. The method requires that

1. **Problem a:** Find the smallest positive root of $f(x) = x - e^{-x} = 0$ using the secant method. Step 1: State the function: $f(x) = x - e^{-x}$.

1. **Problem Q1a:** Calculate absolute error, relative error, and percentage error for each student given true value $25.3267$ and measurements: A = $25.31$, B = $25.33$, C = $25.0

1. **State the problem:** We want to find the square root of 5 using the bisection method with a tolerance of $10^{-2}$. This means we want to find $x$ such that $x^2 = 5$ with an

1. **State the problem:** We want to compute the square root of 3 using the bisection method with a tolerance of $10^{-2}$. The bisection method finds roots of a function by repeat

1. The problem is to determine how many iterations are needed using Newton's Method to approximate a root of a function $f(x)$ with a desired accuracy. 2. Newton's Method formula i

1. The shifting operator $E$ is defined such that $E f(x) = f(x+1)$. For example, if $f(x) = x^2$, then $E f(x) = (x+1)^2$. 2. To find the missing term in Table 1(b), we use the gi

1. O método da bissecção reduz o intervalo de busca pela raiz pela metade a cada iteração. 2. A fórmula para o número mínimo de iterações $n$ para garantir um erro absoluto menor q

1. The problem asks to approximate the root of the function $$f(x) = \tan(\pi x) - 6$$ using the false-position method starting with initial points $$p_0 = 0$$ and $$p_1 = 0.48$$ o

1. **Problem statement:** Solve the system of nonlinear equations: $$y \cos(2xy) + 1 = 0$$