Course: Applied Numerical Ahalysis, ME LU Prof: Hugo Peláez, Student Name: 1. Es
ID: 1868067 • Letter: C
Question
Course: Applied Numerical Ahalysis, ME LU Prof: Hugo Peláez, Student Name: 1. Establish the type of numerical analysis problem according to: (do not calculate) (5 pts) RF-Root Finding, LE-System of Linear Equations, LR-Linear Regression, CF-Curve Fitting. ID differential equation, ND-Numerical Differentiation, NI-Numerical Integration, ODE-Ordinary EV-Eigenvalues. Tip: do not repeat an option. a) The speed of a race car during the first seven seconds of a race is given by t (s) v (miles/h) 0 14 39 69 95 114 129 139 Determine the distance the car traveled during the first six seconds. b) Solve: ? = Vx+ for o rs 5. with y(0) = 0 e) Determine the solution of the equation: log(x) v-3 d) From table on (a) find the speed (v) of the car for t -2.5 seconds e) From (a) what is the acceleration of the car at 5 seconds? 2. True (T) or False (F)? (10 pts) a) Round-off errors increase with the number of calculations performed b) Newton-Raphson method always converge c) Romberg formula is one of the most popular method for numerical integration d) The error of fourth order Runge-Kutta method is greater than the error of Heun method e) First order Runge-Kutta method is equivalent to the Euler method f) A system of 2 ordinary differential equations requires 2 initial conditions to be solved g) Gauss quadrature formula cannot be used for data tables h) System of nonlinear equations can be solved using Gaussian elimination i) Solution of system of nonlinear equations requires to find partial derivatives ) Characteristic polynomial method requires to find a determinantExplanation / Answer
[2]
(a)TRUE
When a calculator or digital computer is used to perform numerical calculations, an unavoidable error, called round-off error, must be considered.This error arises because the arithmetic performed in a machine involves numbers with only a finite number of digits, with the result that many calculations are performed with approximate representations of the actual numbers.
(b)FALSE
Newton's method is guaranteed to converge under certain conditions. One popular set of such conditions is this: if a function has a root and has a non-zero derivative at that root, and it's continuously differentiable in some interval around that root, then there's some neighborhood of the root so that if we pick our starting point in that region, the iterations will converge to the given root.
=>the derivative may be zero at the root; the function may fail to be continuously differentiable; and you may have chosen a bad starting point, one that lies outside the range of guaranteed convergence.
(c)TRUE
Romberg's method is a Newton–Cotes formula it evaluates the integrand at equally spaced points.
(d)FALSE
Heun's method may refer to the improved ormodified Euler's method or a similar two-stageRunge–Kutta method.
=>Runge-kutta fourth order is more accurate than heuns method.
(e)TRUE
(f)TRUE
=>Suppose y1 and y2 are two linearly independent solutions of a second order homogeneous linear equation
y? + p(t) y? + q(t) y = 0.
That is, y1 and y2 both satisfy the equation, and W(y1, y2)(t) ? 0. Then (and only then) their linear combination y = C1 y1 + C2 y2 forms a general solution of the differential equation.
(g)TRUE
(h)FALSE
Gaussian elimination is an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the correspondingmatrix of coefficients.
(I)FALSE
A system of equations where at least one equation is not linear is called a nonlinear system. There are several ways to solve systems of nonlinear equations:
(h)TRUE
When a matrix has elements that are monomials or even polynomials in some set of variables, then its determinant will in general be a polynomial in those variables, and this is sometimes useful in evaluating it.
Thank you.....