9
votes
1answer
86 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
0
votes
1answer
25 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
1
vote
1answer
17 views

Looking for a approximation/solution to my mortgage calculator function

I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
0
votes
1answer
47 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
1
vote
1answer
29 views

Estimating the absolute error of the function $f(x)=4x^2$

I have to estimate the value of $f(x)=4x^2$ for $x\in [1,2]$, and $x$ is unknown. the approximated value for $x$ is $\tilde x$, which is also in $[1,2]$. What is the maximum absolute error of $x$, ...
1
vote
1answer
59 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
1
vote
2answers
129 views

What is the sum that the square root button on calculator does so I can do it without the calculator button [duplicate]

I am not very good when it comes to Maths but the current work I am doing means I need to get better and quick. I have been teaching myself about areas, diagonals and square roots. However I am ...
0
votes
2answers
55 views

Newton Raphson Method for double roots

I am currently working on Newton Raphson Method. I am kind of facing a problem how Newton Raphson Method work on more than second order quadratic functions with double roots. I have googled and ...
15
votes
1answer
199 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
0
votes
1answer
54 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
0
votes
0answers
30 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
1
vote
3answers
48 views

Simplify function with polynomial via least-squares

I want to "adjust" (simplify) $f(x)$, a function, by $g(x)$, a polynomial, via least-squares. I want to write code for that. Apperently my code is issuing wrong results, so I was wondering if my ...
1
vote
1answer
65 views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
2
votes
1answer
59 views

Change of variable

I have to approximate the following integral, using Simpson's Composite $1/3$ Rule: $\displaystyle \int\limits_{0}^1 \mathrm{\frac{e^{2x}}{\sqrt[5]{x^2}}}\,\mathrm{d}x$. The only problem is that ...
0
votes
0answers
30 views

Comparison of trapezoidal , Simpson's 1/3 ,Simpson's 3/8 and Boole's rules.

These rules are often used in numerical integration. How do we analyze the given support points or function and select the most suitable one for best approximation?
1
vote
0answers
38 views

Numerical solution of non-linear differential equation with MATLAB

I need some information to know if I can solve a nonlinear integral equation with terms $ u_{x} $ , $ u_{x}.u_{y} $ , $ u_{xx} $ , $ u_{xy} $ $u_{yy} $ $ u_{x}^{2} $ $ u_{y} ^{2} $ By numerical ...
1
vote
1answer
55 views

Is it possible to calculate $e^x$ given $2^x$?

Given a value $x$, if I have a microprocessor instruction that will give me the value of $2^x$, is it possible to calculate (or approximate) the value of $e^x$ ?
1
vote
0answers
25 views

Initial approximation to inverse of beta distribution function / quantile of beta distribution

I'm interested in implementing an algorithm to find the quantile of the beta distribution, and I'm looking at this paper: Journal of the Royal Statistical Society Series C (Applied Statistics). 1973, ...
3
votes
1answer
74 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
1
vote
1answer
36 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
2
votes
1answer
22 views

How many bits of difference in a relative error?

I would like to know if there is a formula or any other way to find out how many bits of difference between two values given the relative error. For instance: $$\epsilon_{\text{rel}} = \frac{V - ...
2
votes
0answers
28 views

Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme ...
0
votes
1answer
29 views

Absolute error computation

Given is a function $f(x)=4x^{2}$, which we want to evaluate for $x\in \left [ 1,2 \right ]$, $\widetilde{x}\in \left [ 1,2 \right ]$ is the approximation of $x$. What can be the value of the ...
1
vote
0answers
45 views

Approximating an IVP

I wish to solve the IVP: \begin{align} x(0) =& -1 \\ x' =& 1 + x^2 - t^3 \end{align} With a fourth order taylor series method, I solved the ODE on the interval [0, 2] and then made the ...
1
vote
1answer
46 views

How to approximate a smooth function

Now I have a target smooth function f which is infinitely differentiable over $R^d$, $f \in C^{\inf}(R^d)$. $f = \Sigma c_ig_i(x)$, where $c_i$s are unknown coefficients and $g_i(x)$ is a smooth ...
1
vote
0answers
68 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
1
vote
0answers
65 views

Numerically integrating in to Chebyshev polynomial

I'm trying to find the Chebyshev interpolate for an ODE in a given interval. That is, given an ODE that looks something like: $$y'' = g(y) \ y'$$ I want to numerically integrate it inside the ...
1
vote
1answer
119 views

Stable algorithms from a backwards recurrence?

This is homework, so please only hints. We see that the recurrence $$I_n=\frac{1}{n}-\alpha I_{n-1}$$ can be used to compute the value of an integral, but the algorithm is unstable when $|\alpha ...
4
votes
0answers
32 views

How to approximate derivative inside derivative

I am using a box-scheme for solving partial differential equation. The function is approximated with: $$ f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot ...
3
votes
1answer
59 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
1
vote
1answer
71 views

Normal approximation for binomial distribution isn't giving correct result, z score comes out 0

I'm trying to use the normal distribution to calculate approximate values for the (cumulative) binomial distribution with large values (since it's impractical to evaluate the factorials). I'm very ...
1
vote
0answers
71 views

Newton-Rhapson for reciprocal square root

I have a question about using Newton-Rhapson to refine a guess of the reciprocal square root function. The reciprocal square root of $a$ is the number $x$ which satisfies the following equation: ...
1
vote
1answer
26 views

How to determine coefficients of $p(x) = x^6$ with the Chebyshev processing

I want to calculate the coefficients of $p(x) = x^6$ with the Chebyshev processing. How to do that? Following question would be, how to estimate the error in $[-1,1]$, if i only use terms until ...
1
vote
6answers
1k views

How to calculate square root or cube root? [duplicate]

I was reading Richard Feynman biography when I read that one time he was able to calculate the cube root of large number in his brain by just using simple facts of everyday life. So my question is ...
2
votes
1answer
47 views

Approximate $f(x) = \frac{1}{x}$ using $e^x, \sin(x)$ and $\Gamma(x)$

My task is to approximate $f(x) = \frac{1}{x}$ using a linear combination of $e^x, \sin(x)$ and $\Gamma(x)$ in the interval [1, 2] with a step width of 0.01. How is this possible?
1
vote
1answer
137 views

Error analysis of exponential function

By definition: $$ e^x = \lim_{n \rightarrow \infty} ( 1 + \frac{x}{n} ) ^ n$$ I am interesting in calculating the error $$\left | e^x - \left( 1 + \frac{x}{n} \right) ^ n \right|$$ for some fixed $n ...
0
votes
2answers
231 views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm trying to use ...
4
votes
2answers
4k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
4
votes
3answers
360 views

Can I approximate sine and cosine without derivatives?

Assuming I don't know derivatives (and Taylor series) can I manage to approximate sine and cosine of a generic given (rational) angle in radians?
2
votes
1answer
84 views

Brownian bridge distribution: $\sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}}, 0 < a < b <1 $

If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of \begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation} ...
0
votes
0answers
38 views

Approximating the speed of an object from a finite list of approximate positions

An observer watches an object and notes that, for $1 \leq i \leq n$, the object was seen approximately at point $p_i$ in the plane at time $t_i$. For the purpose of this problem, suppose that there ...
1
vote
1answer
202 views

Damped oscillation fit

We have some measurement data like this: The expected behavior of the data is a damped oscillation: $$y=a e^{d*t} cos(\omega t+\phi) + k$$ Where: $t$ Current time $y$ Current deflection ...
0
votes
0answers
37 views

What are the coeff. of this polynomial?

My matlab generates this answer for the problem: p = -20.2090 17.3368 272.9057 -0.7528 Is it correct? It seems I've gotten different results in another ...
1
vote
1answer
46 views

Show that the method is of order 3 if a =-5 and of order 2 if a is not equal to -5

I am studying ODEs and came across this exam question: I have the solution here also: I have been working on this exam question all day and have been stuck for hours. What I don't understand is ...
0
votes
1answer
92 views

truncation error - help

I am trying to understand the concept of local truncation error and came accross this in my lecture notes: what I don't understand here is where the term 'O' comes from and what it stands for in ...
0
votes
1answer
226 views

approximating diagonal of inverse sum of low rank and diagonal matrices

I was wondering if there is any theorem or algorithm to approximate the diagonal elements of the inverse of sum of low rank symmetric positive semi-definite and non-negative diagonal matrix. Let me ...
1
vote
1answer
447 views

How to fit non-linear matlab data?

I'm working on a problem in scientific computing namely fitting data to this equation $c(z) = 4800 + p_1 + p_2 \cdot z/1000 + p_3 \cdot e^{ -p4 \cdot z/1000}$ The data is in a background question ...
1
vote
0answers
62 views

How do you interpolate the local maxima of a set of points in more than 3 dimensions?

I have a set of about 400 points each with 6 coordinates and one scalar value. How can I find out where the local maxima are?
0
votes
1answer
62 views

Approximating a complicated multi-variable function over an interval?

Consider $$ F(\mathbf{r})=F(x,y,z) = \frac{2z^2 - x^2 - y^2}{(x^2+y^2+z^2)^{5/2}} $$ where $x,y,$ and $z, $ are all $n^{\text{th}}$ order polynomial functions of a parameter $t$ with arbitrary ...
3
votes
2answers
146 views

approximate $\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$

I'm trying to find an approximation (or exact solution if possible) for an integral of the form: $$\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$$ I was thinking of somehow applying a Gauss Hermite ...