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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

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Is it possible to restore the missing entry by Newton forward divided difference method?

I've only seen the similiar problem but there are some entries on higher degree given.
Neuer
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Can anyone explain why reducing the stepsize h used in Euler's Method reduces the approximation of a function at a point?

Let $y'=t^{3}y^{2}$ where $y(0)=1$. Approximate $y(1)$ using Euler's method with h=0.25. I learnt online that reducing the step size h reduces the error of the approximation. Can anyone explain why please?
Jnyeboah93
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Euler-Forward product rule

For a numerical approximation we use the Euler-Forward method, we have as definition $$ f'(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ Now we have that $f$ is the product of two other functions namely $g$ and $h$. If we apply the product rule first…
Kaladin
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newton's formula

How is the number of iterations found using the Newton's formula? I tried $|P-P_n|
J.R.
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function program for $e^x$

I'm Using this approximaion: $$e^x\approx \text{myFunc}(x)=\sum_{i=0}^{10}\frac{x^i}{i!}$$ I'm trying to write a function program to evaluate $e^x$ and have an error that is $\leq$ $10^{-7}$ which is the taylor approximation for $e^x$ up to 10. Is…
user179523
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A proof of Newton's Iteration for Finding Square Roots

I find this theorem in my textbook: Assume that $A>0$ is a real number and let $p_0>0$ be an initial approximation to $\sqrt{A}$. Define the sequence $\{p_k\}_{k=0}^\infty$ using the recursive rule $p_k=\frac{p_{k-1}+\frac{A}{p_{k-1}}}{2}$, for…
Jiabin He
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First-degree spline interpolation problem

I have been spending hours working on a problem from Cheney & Kincaid on first-degree spline interpolation problem but am heading nowhere. Any explanation, pointers or hints from you would be bery much appreciated: Defind a function $g$ by the…
A.Magnus
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Expanded form of Divided differences

I am studing numerical analysis and I'm reading In wikipidia about it. I want to understand the Expanded form - I tried to prove this($f[x_0,\ldots,x_n]=\ldots$) by induction and failed. Any Ideas on how this can be proven ?
Belgi
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how to prove the eigenvalues of tridiagonal matrix

Assume the tridiagonal matrix $T$ in this form: $$ T = \begin{bmatrix} a & c & & & \\ b & a & c & & \\ & b & a & \ddots & \\ & …
user157745
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For what values of c will the iteration $x_{n+1} = g_c(x_n)$ converge to $\alpha_c$

Consider the equaction $x=g_c(x)\equiv cx(1-x)$, with c a nonzero constant. This equation has two solutions, and we let $\alpha _c $ denote the nonzero solution. What is $\alpha _c$?For what values of c will the iteration $x_{n+1} = g_c(x_n)$…
user124557
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functional iteration

This is a question from D Kincaid & W Cheney, Numerical Analysis (3ed), Brooks-Cole 2002; Find the condition on $\alpha$ to ensure that the iteration will converge linearly to a zero of $f$ if started near the zero. This question is in the Newton's…
aranel
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show the function has exactly one root

Given: Function: $f(x)=x^2-2x-3$ $[1,4]$ Question: Show that the function has exactly one root in $(1,4)$ My Answer: The function $f(x) = x^2-2x-3$ has one root in [1,4] $f(1)=1^2-2(1)-3=1-2-3=-4$ $f(4)=4^2-2(4)-3=16-8-3=5$ Since $f$ is continuous,…
chris
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Graphical estimate of convergence rates?

I am studying some numerical optimization methods, but I am not an expert in numerical maths. Question: If the convergence rate is linear, then the logarithm log(x_n) of the distance x_n to the minimum plotted against n gives a straight line. Now…
Karl
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Is there a numerical method to restore symmetrizability and row-stochasticity in a matrix?

I have a radiation-transfer matrix F that had been computed using some raytracing Monte Carlo method. From physics it is clear that it must be row-stochastic :$\sum_{j=1}^{N}F_{ij}=1$ because energy cannot be lost. It is also clear that it must be…
george heitzer
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