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January 2016 - Stony Brook University
January 2016 - Stony Brook University

... b) (4 points) Describe the algorithm of Newton’s method for solving the nonlinear equation ∇L(x, y, λ) = 0 to obtain the critical points (x∗ , y∗ , λ∗ ). Show one step of the Newton’s method the initial guess (x0 = 1, y0 = 2, λ0 = 0). c) (3 points) Is (x∗ , y∗ , λ∗ ) a minimum, maximum, or saddle po ...
Here
Here

... using a quadratic in x as the approximating function. b) Solve the problem by collocation, setting the residual to zero at x = 0.5. c) Solve the problem by Galerkin’s method. 5. Develop the elements equations for a 10-cm rod with boundary conditions of T(0, t) = 40 and T(10, t) = 100 and a uniform h ...
§3.1 Introduction / Newton-Cotes / The Trapezium Rule
§3.1 Introduction / Newton-Cotes / The Trapezium Rule

... Given a real-valued function f that is continuous on [a, b], can we find an estimate for Z b ...
Statistical Computing and Simulation
Statistical Computing and Simulation

... could count “nlminb” as one of the method (for replacing Newton’s method). Also, similar to what we saw in the class, discuss what is the influence of starting points to the number of iterations. Experiment with as many variance reduction techniques as you can think of to apply the problem of evalua ...
method - csc113ksu
method - csc113ksu

... The Method toString • Implementing toString method in java is done by overriding the Object’s toString method. • The java toString() method is used when we need a string representation of an object. • It is defined in Object class. • This method can be overridden to customize the String representat ...
1-chapter1-1_objects-and-methods
1-chapter1-1_objects-and-methods

... CSC 113: Computer programming II ...
Computerised Mathematical Methods in Engineering
Computerised Mathematical Methods in Engineering

... This means that all ci,j are determined by values inside the boundaries Initial condition [e.g. c(x,0)=sin(xπ)] must satisfy boundary conditions [e.g. c(0,t)=0] ...
Practice Final
Practice Final

... 7. Find a, b, c such that y = a2 + b cos 2x + c sin 3x is the least square approximation to y = x in [−π, π] with respect to the weight function w(x) = 1. 8. Derive the nonlinear system for a, b such that the exponential function y = beax fit the following data points, (1.0, 1.0), (1.2, 1.4), (1.5, ...
Chapter 4 part 2 Runge
Chapter 4 part 2 Runge

... One problem with the Muskingum method is that it assumes that the storage equation is linear with depth. Although this simplifies calculation considerably, it doesn’t necessarily have anything to do with reality. A more accurate method would allow for the storage function to be any function of depth ...
A proposal of variant of BiCGSafe method based on optimized
A proposal of variant of BiCGSafe method based on optimized

... of combination of two polynomials was generalized as a form of product of two polynomials in 1997. However, the optimization of product of polynomials remains as an open problem. The first solution among neive realization of product-type iterative methods was made partly by Fujino et al. in 2005 owi ...
3 Approximating a function by a Taylor series
3 Approximating a function by a Taylor series

... What does this mean for computation? In nearly all our computations, we will replace exact formulations with approximations. For example, we approximate continuous quantities with discrete quantities, we are limited in size by floating point representation of numbers which often necessitates roundin ...
The calculation of the degree of an approximate greatest common
The calculation of the degree of an approximate greatest common

... The calculation of the degree of an approximate greatest common divisor (AGCD) of two inexact polynomials f (y) and g(y) is a non-trivial computation because it reduces to the estimation of the rank loss of a resultant matrix R(f, g). This computation is usually performed by placing a threshold on t ...
Integration Formulas
Integration Formulas

... Use Newton’s method with initial guess of x1 = 1 to approximate a zero of f (x)  x 3  x 2  1 . ...
BBA IInd SEMESTER EXAMINATION 2008-09
BBA IInd SEMESTER EXAMINATION 2008-09

... Note: Attempt six questions in all. Q. No. 1 is compulsory. ...
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Root-finding algorithm

A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. Such an x is called a root of the function f.This article is concerned with finding scalar, real or complex roots, approximated as floating point numbers. Finding integer roots or exact algebraic roots are separate problems, whose algorithms have little in common with those discussed here. (See: Diophantine equation for integer roots)Finding a root of f(x) − g(x) = 0 is the same as solving the equation f(x) = g(x). Here, x is called the unknown in the equation. Conversely, any equation can take the canonical form f(x) = 0, so equation solving is the same thing as computing (or finding) a root of a function.Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards a limit, which is a root. The first values of this series are initial guesses. Many methods computes subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points.The behaviour of root-finding algorithms is studied in numerical analysis. Algorithms perform best when they take advantage of known characteristics of the given function. Thus an algorithm to find isolated real roots of a low-degree polynomial in one variable may bear little resemblance to an algorithm for complex roots of a ""black-box"" function which is not even known to be differentiable. Questions include ability to separate close roots, robustness against failures of continuity and differentiability, reliability despite inevitable numerical errors, and rate of convergence.
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