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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

14158 questions
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Iteration convergence.

How can I solve this problem? Let $$x(n+1)=-\frac{\exp(x(n)/2)}{5}$$ be a given sequence. Prove using the Banach contraction principle that this sequence converges to some fixed point $X$ with $x(0)$ in some interval $[a,0]$ where $a<-1/5$.
Seany Adams
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Converting a PDE into a matrix equation of the form Ax=b and discuss error estimate using discretization method

Let $$-u_{xx}+u=f(x)=(\pi-x)\sin x, \ x\in (0,\pi),\ u(0)=0,\ u(\pi)=2.$$ Solve the problem on $$h=\frac{\pi}{N}, N=16,32, 64, 128, 256$$ $$0=x_{o}
S. Phanzu
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Simpson's rule in numerical methods

In the following code I have implemented composite Simpson's rule. However I should be getting approximately $291$ but for some reason I am getting something different. I implemented a few other methods to test it and the proper answer was $291$ So…
fr14
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An undefined index in Kress's Numerical Analysis

I am reading Kress's Numerical Analysis. There is a paragraph on page 84 which is discussing what will happen if we "perturb" an equation $Ax=y$. It says: If for some $\delta\in \Bbb{C}$ we perturb the right-hand side by setting…
bfhaha
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Find the four points $b \in \mathbb{R}^2$ such that $||b||_2=1$ and $||Ab||_2=1$

Define a matrix $A=\begin{bmatrix} 3 & -1 \\1 & 0 \end{bmatrix}$ which represents a linear transformation in $\mathbb{R}^2$. Consider the following norms on $\mathbb{R}^2$: $||(x,y)||_p=(x^p+y^p)^{1/p}\\ ||(x,y)||_\infty=max{|x|,|y|}$ Find the four…
Sarah2018
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$ p^{*} $ must approximate $ p $ with relative error at most $ 10^{-3} $

The book Numerical Analysis by Richard L. Burden has the following exercise: Suppose $ p^{*} $ must approximate $ p $ with relative error at most $ 10^{-3} $. Find the largest interval in which $ p^{*} $ must lie for each value of $ p $. a) 150…
miguel
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Is there any algorithm that finds all zeros(roots) of a continuous function $F:\mathbb R^n\to\mathbb R^n$?

Is there any algorithm that finds all zeros(roots) of a continuous function $F:\mathbb R^n\to\mathbb R^n$? I've seen some algorithms that find one zero, such as Newton's method. But I need to find all zeros (I need this because I want my computer to…
zxcv
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How to round down negative numbers?

When discussing question on another site, I came to a disagreement about rounding down negative numbers. The two positions were: Rounding down is always toward negative infinity. It is ambiguous. It is defined as both rounding toward negative…
ravery
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$A$ can be written as the sum of rank-one matrices, i.e $A = \sum _{i=1}^r \sigma_i u_i v_i^*$

The singular value decomposition of a complex-values $n\times m$ matrix $A$ is defined as $$A = U \Sigma V^*$$ where $U$ and $V$ are unitary $n \times n$ matrices and $m \times m$ matrices respectively, and $Σ$ is an $n \times m$ rectangular…
user104
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Is GMRES an iterative or a direct solver?

Given is a system of linear equations $Ax=b$ and $A$ is of size n-by-n. If we use the generalized minimal residual method (GMRES) we iteratively calculate approximate solutions for Krylov subspaces of increasing dimension. Usually we abort…
koalo
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Why does the following method work in isolating pro-numerals?

A question simply asks to isolate (x) from 5xy=c, which is simple, right? Naturally, you'd divide both sides by (5y). But, I was wondering why does that work? 5xy = xy + xy + xy + xy + xy and 5y = y + y + y + y + y? Thus, if you were to divide…
Dr Xionel
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Numerical Analysis. Newton-Raphson formula.

Attempt: a) $g(x) = x- f(x)/f’(x) = x – [(x-2)^4]/[4(x-2)^3] = (2-x)/4+x = (3x+2)/4$ So, $p_k = [3p_{k-1}+2] /4$ b) p(1)=2.1; for j=2:5, p(j) = (3*p(j-1)+2)/4 end p = 2.1000 2.0750 2.0563 2.0422 2.0316 So,…
John Lennon
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Numerical Analysis. Bisection method.

What will happen if the bisection method is used with the function $f(x) = \tan(x)$ and a) $[3,4]$ b) $[1,3]$ Attempt: Check the signs of the function: $f(x) = \tan(x)$ a) $f(3)f(4) = -0.165 <0 \implies$ the root is between $3$ and $4$ b)…
John Lennon
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In Dekker's and Brent's Method, in the initial steps, it states that $a$ and $b$ must be swapped if $|f(a)| < |f(b)|$, why is this step being done?

In Dekker's and Brent's method, in the initial steps, if $|f(a)|<|f(b)|$, we swap $a$ and $b$. Why is this? I've searched for reasons why, but I cannot find a reason why. Cleve Moler mentions: $b$ is the best zero so far, in the sense that $f(b)$…
Moffee
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Loss of significance error

Give exact ways of avoiding loss-of-significance errors in the following computations: a. $log(x+1)-log$, with large $x$ b. $\frac{1-cosx}{x^2}$, with $x\approx 0$ c.$(1+x)^{1/3}-1$, with $x\approx 0$ Am I doing this correctly? a. I got…
Lays
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