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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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Numerical methods to find the root of a continous function from $R$ to $R^2$

I have a continuous function from $R$ to $R^2$, which I want to find a root of. I tried finding a root of the sum of squares, but since that function is always positive I can't use bisection, and any other method I tried didn't converge. What root…
Command Master
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Suppose $\varphi \colon [0,1] \to \Bbb R$ is three times continuously differentiable function

I was looking at old exam papers and was stuck on the following problem: Suppose $\,\,\varphi \colon [0,1] \to \Bbb R$ is three times continuously differentiable function. Suppose further that the iterates defined by $x_{n+1}=\varphi(x_n), n…
user52976
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What's the diffrence between Secant method and False position method?

I've been studying numerical analysis and I ran into false position method and I can't figure out its diffrence between secant method , because I also heard that the convergence is guaranteed in false position method but not in secant method. And…
Elias
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Understanding why Roots of Chebyshev Polynomial Give best Interpolation

So I know that if we choose the nodes of the interpolating polynomial to be the roots of the Chebyshev polynomial then the error is minimized. My textbook doesn't fully prove this, they just give a sketch of the proof. In the sketch they mention…
k12345
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Some questions about variations of fixed point method

I'm doing some excercises in Fixed Point Iteration methods with Matlab. I have to find roots for $f(x)=e^x -x -1.9\cos x$ by using $x_{n+1}=g(x_n)$. I know how to choose $g(x)$ such that I can find both roots. The following part of the exercise is…
Applied mathematician
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Solving for (or approximating) $x$ in $\frac{x-\sin x}{x^{0.4}} = c$

I'm an engineer and I'm deducting a formula, and I found a BIG problem: I can't isolate $x$. The formula is: $$\frac{x-\sin x}{x^{0.4}} = c$$ where $c$ is a constant. How to find the value for $x$, or at least a good approximation?
Rafael Araujo
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Loss of "significance" in $x-y$, $x=0.3721478693, y =0.3720230572$

If we perform the calculations in a computer with a five digit mantissa then I get: $$\begin{aligned} \mathrm{fl}(x) &=0.37215 \\ \mathrm{fl}(y) &=0.37202 \\ \mathrm{fl}(x)-\mathrm{fl}(y) &=0.00013 \end{aligned}$$ The exact calculation are:…
Sorry
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Choosing the initial values (Newton's method)

Let, say, I want to find an approximation of the root $r$ for $x^{3}=x+1$ using Newton's method. $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \quad(n \geq 0)$$ I guess the initial point is denoted by $x_0$. My question…
Sorry
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What are some good Numerical Analysis textbooks?

I have been doing some research lately about numerical analysis textbooks, but I have not found anything successfully. I was wondering if there is like some sort of standard, for example, whenever someone mentions Calculus, I would think of Spivak…
paul
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Newton (Iterative) Vs. Babylonian (Direct) For Roots

Is Newton's iterative method for finding a square root more efficient then the Babylonian method? Considering most roots are irrational, which method would get me within, say 16 decimal places, the fastest? I need a high level of precision so I was…
MarJamRob
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How to solve this problem of Numerical Differentiation

I have a doubt on how to solve this problem, I don't know what does it mean. Calculate two approximations by differentiation of first order with different truncation error and compare them with the analytic solution of the following given functions…
Iesus Dave Sanz
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Newton's divided differences expanded formula

For the proof of the expanded divided differences formula, I learned how to use induction. Expanded form: \begin{aligned}f[x_{0}]&=f(x_{0})\\f[x_{0},x_{1}]&={\frac {f(x_{0})}{(x_{0}-x_{1})}}+{\frac…
Hoda Bibo
  • 405
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When is Newton's Method guaranteed to converge to a good solution (non-linear system)?

My knowledge of Newton's Method is partial. I am trying to understand what guarantees this method can give on the solution of systems of non-linear equations. Specifically, I have a set of non-linear equations that are easily twice differentiable.…
Bitwise
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Linear equation with prescribed precision of result

Let $x$ be an unknown real vector of size n. Suppose we can find n vectors $v_i$, and are given the values $x^Tv_i$. Then we can simply solve for $x$ by Gauss or some other method and determine $x$ uniquely and precisely( iff $v_i$ are lin…
wood
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How to find the limit of a convergent matrix?

I'm trying to learn how to show a series of matrices is convergent and find the limit. However my numerical analysis books fail to mention this and I cannot find any relevant material online! Anyway say I had $\sum_{k=0}^ \infty A^K$, where A =…
user67411
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