Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

14158 questions
0
votes
1 answer

Numerical Integration Proof

I have been working on this problem for 3 days and I still have no idea where to start with it. I previously proved an equation: $$hf'(x) = \frac{1}{2}(f(x+h)-f(x-h)) - \frac{h^2}{12}(f''(x+h)-f''(x-h)) + O(h^5)$$ and I am asked to prove that,…
user728757
  • 3
  • 4
0
votes
1 answer

Numerical Analysis - Synthetic Analysis

I have been working on this homework problem for 2 hours, having thought it was pretty simple, and its really annoying me. I am given two operators and a function f(x) that is five times differentiable on some interval. $$\alpha f(x)=f(x+h/2) -…
user717258
0
votes
2 answers

Cubic Converging Functions with Newton

I have been tasked with attempting to find properties with the function f(x) that would make It such that using newton's method would converge to a particular root at least cubically. I don't exactly understand what this is asking me because I'm not…
John Foe
  • 1
0
votes
1 answer

Numerical method for solving Fredholm integral equation

I learned numerical integration methods such as rectangualr, trapezoid, Simpson, Roberg Integration, Richardson Extrapolation, Adaptive Quadrature, Gauss Quadrature. My professor ask students to solve the Fredholm integral equation by using the…
alryosha
  • 563
0
votes
1 answer

Condition number for 2-norm?

enter image description here Can anyone tell me if I have made right thoughts in the attached task? To solve the system I just solve $x=A^{-1}b$. To find the 2-norm of matrix $A$ I use the "power method", because the 2-norm for a normal matrix like…
user710461
0
votes
1 answer

Project e^x using the monomial and legendre basis

Project $e^x$ onto $P_2$ with the $L^2$ inner product on $[-1,1]$ using: a) The monomial basis {$1, x, x^2$}. b) The Legendre basis {$1,x,\frac{1}{2}(3x^2 - 1)$}.
Vishnu P
  • 43
0
votes
1 answer

Prove the matrix A is singular with the conditions below.

Let $A=[a_{i, j}]$ be an $n\times n$ matrix with real entries. spz that there is an $m$ with $a_{i, j} =0$ for $i\ge m$, $j\le m$ and $a_{i, i}$ not equal to zero for $1\le i
Sherien Hassan
  • 21
0
votes
1 answer

Newton and Secant Method approximate roots is a convergence sequence

Question: Prove that if function has a root in [a,b] and no critical or inflection points in the interval, then the sequence of newton method and secant method approximate roots is a convergent sequence. I know that the only time newtons method…
Sherien Hassan
  • 21
0
votes
1 answer

About fixed point iterations

I've read that to insure that a function $g(x)\in C[a,b]$ has a unique fixed point ,is to prove that the absolute values of its derivative $g^{'}(x); x\in]a,b[$ must be less than $1$. what does this concept mean in analysis?
soso sos
  • 349
0
votes
1 answer

Deriving error for second derivative approximation

Using Newton's difference form $f''(x_0) \approx 2f[x_{-1} x_0 x_1]$ as an approximation, I need to show that this approximation is first order accurate. So WTS $f''(x_0) - P''_n(x_0)$ is first order accurate. Using the error formula, we have…
Lutterbach
  • 627
0
votes
2 answers

Runge-Kutta method $K_1$ term in $K_2$ equation

I'm studying Runge-Kutta method and I don't get some part of it. Runge-Kutta method is as follows \begin{equation} x(t+h) = x(t) + w_1 K_1 + w_2 K_2 \\ K_1 = h(f, x) \\ K_2 = hf(t+\alpha h, x + \beta K_1) \end{equation} In above K2 equation, K1 is…
alryosha
  • 563
0
votes
3 answers

Using double precision, what is ((0.5 + 1e-19) - 0.5) * 1e19?

Using double precision, what is ((0.5 + 1e-19) - 0.5) * 1e19? Should it be 0 or 1? Why? EDIT: Matlab is giving me 0, but I was expecting 1. Why? Shouldn't 0.5 + 1e-19 produce a submornal number that is greater than 0? > >> ((0.5 + 1e-19) - 0.5) *…
user1559897
  • 1,807
0
votes
0 answers

What is the interpretation of backward stability?

The book I am reading by Trefethen and Bau defines backward statbility in the following way... A problem is backward stable if for each x, $$ \tilde{f}(x) = f(x) $$ for some $\tilde{x}$ with $\frac{||\tilde{x} - x||}{||x||} = O(\epsilon)$. It…
user1559897
  • 1,807
0
votes
0 answers

H.M or G.M in Bisection method

In bisection method we use A.M mean to bisect the interval and to find the approximate root for the equation $f(x)=0$ can we use the H.M or G.M instead of A.M? If yes what are the consequences?
kam
  • 151
0
votes
1 answer

If $\Delta^n0^m=\Delta^nx^m|_{x=0}$ , then what is the value of $\Delta^30^6$?

Its a numerical analysis question where $\Delta$ is the forward difference operator. But although I have learned about the different operators I really can't understand the meaning of $0^m$. The answer options are (a) 500 (b)515 (c)530 (d)540 I am…
Azka
  • 3
1 2 3
…
99 100