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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

Computing Values of Functions

Postings: 232

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly out output.

Ordinary Differential Equations

Postings: 126

An ordinary differential equation is an equation containing a function of one independent variable and its derivatives.

Partial Differential Equations

Postings: 87

A partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives.

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

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

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. 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

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

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

The 1D heat equation

Please see the attached file for the fully formatted problems. Solve the heat equation u_t = ku_xx for 0<x<L With boundary conditions u(0,t)=u(L,t)=0 Solve for the initial value conditions: a. u(x,0) = sin(5*Pi*x/L) b. u(x,0) = x c. For part b, plot the solution at t=0, 0.1, 1

Cancellation Round-off Error : Numerical Analysis : Confirmation

I have numerically solved the following quadratic equation: 1.002x2 - 11.01x + 0.01265 = 0. IS IT POSSIBLE THAT IN THIS INSTANCE EQUATION (2) EQUALS (2A) BELOW: If b2 - 4ac >0, the quadratic equation ax2 + bx +c = zero has two real solutions x1, x2 given by the typical: (1) x1 = (-b + sqrt(b^2-4ac))/ (2a)

Actual, Absolute and Maximum Relative Errors in Data

A touring company has the following box office takings for the recent month. Venue Clarion West Riding West End Burlington Huntingdon Erie Adelphia Wilson Reading How do I estimate the abosolute and relative maximum and average errors in the total of the above rounded data? How do I determine the actual error i

PDE : Riemann's Method for Solving Cauchy Problem

Hello. Thanks for help! I will use * to indicate a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. This is the probelm: Use Riemann's method to solve the Cauchy problem: u*xx + 4u*xy +3u*yy = 1, u=1 and u*n = square root of 5 times x, on the intial curve y=2x. If this

Heat Equation, Boundary & Steady State Conditions, Initial Value

Please help with the following problems involving numerical analysis. (a) Find all the separated solutions of the attached heat equation (satisfying the attached boundary condition) (b) Use these separated solutions to write a series solution for the initial value problem posed by the attached pde and the attached boundar

Initial Value Problem (IVP); Euler's Method; Step Sizes

4. Consider the initial value problem (IVP): y'(t) = 3+t+y y(0)=1 a) Approximate y(1) using Euler's method and step sizes of 0.2. Perform these calculations by hand. What is the exact value of y(1)? b) Use the computer (e.g. ODE Architect, ODE Toolkit, or your own program) to approximate y(1) using step sizes of 0.1, 0.05,

Bisection and newton's method

Q Using the bisection method, find the positive root of 2x(1 + x^2)^-1 = arctan x. Using this root as x0; apply Newton's method to the function f(x) = arctan x: Interpret the results you obtain.

Solve an IVP ODE using the Method of Variation of Parameters

Please see the attached file for the fully formatted problems. Solve an IVP ODE using the method of variation of parameters Find the solution of the system X' using the method of variation of parameters 2 0 0 cos(t) X' = -1 0 -1 X + sin(t) 1 1 2 e^-t that satisfies the int