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# Numerical Analysis

Numerical analysis is the study of algorithms that use numerical approximations for the problems of mathematical analysis. It continues this long tradition of practical mathematical calculations. Modern numerical analysis does not seek exact answers. This is due to the fact exact answers are often impossible to obtain in practice. Instead, numerical analysis is concern with obtaining approximate solutions while maintaining reasonable bounds on errors.

Direct methods compute the solution to a problem in a finite number of steps. These methods give precise answers if they are performed in infinite precision arithmetic. Examples of direct methods include Gaussian elimination, the QR factorization methods for solving systems of linear equations, and the simple method of linear programming. Finite precision is used and the result is an approximation of the true solution.

Iterative methods are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the imit. A convergence test is specified in order to decide when a sufficiently accurate solution has been found. Even using infinite precision arithmetic, these methods will not reach the solution within a finite number of steps. Examples of the iterative methods include Newton’s method, the bisection method and Jacobi iteration. Iteration methods are generally needed for large scale problems.

Since the advent of computers, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems. Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results.

## Categories within Numerical Analysis

### Compute the fourth root of 20 using Newton-Raphson interation

Give the Newton-Raphson iteration scheme for computing the pth power of a number. Specialize this to computing the fourth root of 20. You can take the initial guess to be 2. At each iteration, state the number of correct digits.

### Draw and Sketch Velocity

Answer the following questions about shipping lobsters. For each problem: ★ Include the equations or formulas you used. ★ Explain, in words or with mathematical steps, how you arrived at your answers. You do not have to submit sketches, but you may find that drawing parts of the problem on scratch paper can help you unders

### Quantitative Problem Solving with excel

Question 6: 15 points The biggest inventory problem at the Barko facility is the storage of boom sections for their various Knuckleboom models. There are two types of boom sections: short and long. The following table outlines the demand in the next 5 months and the projected purchase price for each type of boom section. De

### Solving 7 questions on various topics

Problem 1 Suppose a manufacturing company makes a certain item. The time to produce each item is normally distributed around a mean of 157 minutes with a standard deviation of 46 minutes. (a) What proportion of the items will take more than 2 hours to make? (b) What proportion of the items will take between 160 and 200 minu

### Drawing phase diagrams

Please explain the steps and show the graphs. Thank you.

### Solving Discrete Mathematics Questions

1. After a weekend at the Mohegan Sun Casino, Gary finds that he has won \$1020—in \$20 and \$50 chips. If he has more \$50 chips than \$20 chips, how many chips of each denomination could he possibly have? 2. If there are 2187 functions f : A→B and |B| = 3, what is |A|? 3. Let A ⊆ {1, 2, 3, . . . , 25} where |A| = 9.

### Separation of Variables, System Progress, and Phase Diagrams

(1) Solve by separation of variables and solution to the ODE (see attached file for formula). (2) Solve the equation (see attached file for formula). (3) For the given graph, re-sketch a graph for x01 and indicate the system progress from x01 as t --> infinity. Identify any critical points as stable, unstable, and semi-stabl

### Graphing and Harvesting Functions

See the attached file. Harvesting: x = a1*x - a2*x^2 - h a1 > 0, a2 > 0, h>0 1. |x| = tons and |x| = tons/years What are the dimensions of a1, a2, and h? (see attached file for more details) 2. Use the quadratic formula to show that the phase diagrams (x vs. x)have no intercepts: (see attached file for

### Identifying and Recognizing Variations in Sets

I am currently working on sets and how to identify/recognize variations based on symbols. Is there any kind of "outline" or list/reference that I can utilize to better understand their core meaning? A' = B' = element of... subset of...

### Finding the "Divisibility Rule" for Number 16

Problem: A number is divisible by 2 if the last digit is divisible by 2. A number is divisible by 4 if the last two digits form a number divisible by 4. A number is divisible by 8 if the last three digits for a number divisible by 8. Objective: Find a possible pattern. Use this pattern to describe a "divisibility rule" fo

### 10 Differential Equation Questions

Use separation of variables to solve to the ODE: N ̇=〖{k〗_+-k_- ln⁡〖(N)}N〗;k_+,k_- and t>0;N(0)= N_0>0. Hint: Use u = ln(N) for u-substitution. Substitute your solution into the differential equation and show that it is in fact the solution. Evaluate your solution for N(t) as t→∞ . For what va

### Springdale Shopping Survey

Please see attached Excel sheet. Also below is the link to the free book for reference. (I would start there!) The question is on Page #308. http://www.xn--klker-kva.hu/wp-content/uploads/2013/03/Introduction-to-Business-Statistics-7th-Edition-2008.pdf 1. Item C in the description of the data collection instrume

### Solution of differential equation on interval of x values

Determine the order of each differential equation, and whether or not the given functions are solutions of that equation on some interval of x values. (a) (y')^2 = 4y; Y1 = X^2, Y2 = 2X^2 , Y3 = e-x (b) Y" +9y =0; Y1 = 4sin3X, Y2 = 6sin(3X+2)

### Analysing Frequency Distributions

Using Excel, prepare a frequency distribution from the data collected (see attached below). Calculate the Standard Deviation of your data. Is this a normal distribution? Every day I leave my house 15 min before I need to be to work, what are the chances I will be late to work on any given day? There are several differe

### Two-Dimensional Wave Equation

1. Find the solution to the two-dimensional wave equation [see the attachment for the full equation] 2. Solve the two-dimensional wave equation for a quarter-circular membrane [see the attachment for the full equation] The boundary condition is such that u=0 on the entire boundary. 3. Consider Laplace's equation [see t

### Numerical analysis and 7th degree splines

Given the points x={xo, x1, x2,.... xn}^T and the function values f={fo, f1, f2, ....fn}^T at those points, we want to generate a 7th degree spline, i.e. a piecewise 7th degree polynomial approximation. a) Why would we want to do this? Why not just use Newton's Interpolatory Divided Difference formula to get an nth degree in

### Interval of existence of an IVP

Please solve the following problem using the step by step method: Find the unique solution (and proof that it is unique) and the maximal interval of the initial value problem x' = x^3, x(0) = xo for any xo E R.

### The Flow Invariant for the Nonlinear System

Please solve the following numerical analysis problem: Determine the flow Qt : R^2 into R^2 for the nonlinear system: x' =f(x) with f(x) = [ -x1 ] [ x1^2 + 2x2 ] and show that the set S = { x E R^2l x2 = -x^2/4 } is invariant with respect to the flow {Qt}. Plea

### Fixed point iteration, unique solution converge

Show that for any constants c and d, |d| < 1, the equation x = c + d cos (x) = g(x) has a unique solution alpha. In addition, show that the iteration x_n+1 = c + d cos (x_n) will converge to alpha. Bound the rate of convergence.

### Determining the Height an Arrow Flies

An arrow of mass m is shot vertically upward with initial velocity 160 ft/sec. it experiences both the deceleration of gravity and a drag force of magnitude kv^2 due to air resistance, where v is the velocity and k is a coefficient of air resistance. Assuming a drag coefficient of k=1/800 lb m/ft, how high does the arrow go? doe

### Deriving the Black - Schole Equation

The BSM equation for a derivative F on a stock paying no dividends is: -See attachment If you were to transform this PDE with the substitution S = Ke^x, where x is a variable and K a constant, what would be the resulting equation?

### Proof in Numerical Analysis: Taylor's Theorem

Suppose that y(x)' : f(x,y(x)) on the interval [x0, x1] with y (x0) = y0. Assume that a unique solution y exists such that it and all of its derivatives up to and including the third order are defined and continuous on [x0, x1]. Using Taylor's Theorem (and the Mean Value Theorem, if necessary) prove that... See the attached f

### Proof in Numerical Analysis

Use the Secant method (defined in the book) to show that sequence below converges to (square root Q), where Q > 0, given "good" starting values x_0 and x_1: x_n+1 = (x_n x_n-1 + Q) / (x_n + x_n-1). Come up with a similar recursion for calculating Q^(1/3) using the secant method.

### Rates of Convergence of Powers

1. Suppose that 0 < q < p and that alpha_n = alpha + O(n^-p). Show that alpha_n = alpha + O(n^-q). 2. Make a table listing h, h^2, h^3, and h^4 for h = 0.5, 0.1, 0.01, and 0.001, and discuss the varying rates of convergence of these powers of h as h approaches zero. Please provide a brief text description explaining the st

### Separation of Variables for PDEs

Hello. I am having some trouble with the following PDE: Laplacian(u(x,y,z)) = u(x,y,z) * (-2*E/h^2)*(1 + (GM/(2(x^2+y^2+z^2)^(1/2))))^4 Where, G,M,E, and h are all constants. The problem that I'm having is that there is a nonconstant factor of (x^2+y^2+z^2)^(-1/2) appearing on the RHS of this equation, making it non-tri

### Growth Model Populations

The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P0 has doubled in 5 years, how long will it take to triple? To quadruple?

### Non-Trivial Solution

PLEASE SEE ATTACHMENTS. Solve question #3

### Laplacian

(1) Show u belongs to C^1(U), and calculate ux, uy. (2) Calculate uxx, uyy, uxy and show that uxx and uyy are continuous in the whole disk, but u does not belong to C^2. Calculate f := - delta u. Please see the attached image for complete question.

### Find y as a Function of x if...

Find y as a function of x if... 1a) (x^2)y" - 5xy' - 27y = x^7 w/ y(1) = 4, y'(1) = 6 1b) y'''-11 y'' + 18 y' = 0 w/ y(0) = 1, y'(0) = 3, y"(0) = 4 1c) y''' +49 y' = 0 w/ y(0) = -6, y'(0) = -42, y''(0) = 147 1d) y^(4) - 4 y''' + 4 y'' = 0 w/ y(0) = 1, y'(0) = 4, y''(0) = 4, y'''(0)=0 1e) y''' -2 y'

### Encompassing initial value problems

Solve the initial value problems; 1a) t(dy/dt) + 5y = 7t w/ y(1) = 3 1b) (dy/dt) + 0.1ty = 7t w/ y(0) = 4 1c) 11(t+1)(dy/dt) - 7y = 28t for t > -1 w/ y(0) = 20 1d) (dy/dt) + 2y = 25sin(t) + 30cos(t) y(0) = 1 1e) (dy/dt) + 5y = cos(4t) w/ y(0) = 0 1f) 9(dy/dt) + y = 18t w/ y(0) = 1 find y