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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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Implicit function - Finding the $y=f(x)$ for each $x$ base on Newton-Raphson

How could I find the $y=f(x)$ for each $x$ in an implicit function base on Newton-Raphson ? For example - Given the implicit function - $$3x^7+2y^5-x^3+y^3-3=0 $$ how could I find the $f(0)$ base on Newton-Raphson ? Any idea or hint would be…
URL87
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Finding Complex Eigenvalues of Hessenberg Matrix

Given a Hessenberg matrix, i wish to compute its eigenvalues using QR Algorithm. The problem is that the matrix has complex eigenvalues and my implementation of the QR Algorithm can't find them. It sucessfully finds the eigenvalues that have only…
arthurp36
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Fixed Point Iteration Convergence Requirements

I understand that to use Fixed Point Iteration we demand that the function will be continues and will both the range and domain will be in the same closed interval and that $|f'(x)|<1$ for all $x$ in the interval. Do we demand that the function have…
newhere
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how can I solve a equation like that $40939=200 \cdot 0.54 \cdot T+\sigma \cdot \varepsilon\cdot T^4$ with a simple scientific calculator?

$\varepsilon = 0.8$ and $\sigma = 5.669 \cdot 10^{-8}$ We are only allowed to use simple scientific calculador during the tests. How can I solve this equation like that? Find T. $40939=200 \cdot 0.54 \cdot T+\sigma \cdot \varepsilon\cdot T^4$
Matheus José
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How to find the least positive root?

How to numerically find the least positive root of equation? What numerical method should I use and with what starting conditions? I believe I don't have all the roots to be sure, do I? Although answer to general case would be nice, I would also add…
zduny
  • 139
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How to convert 2nd order 2d equation to to first order

There is the equation for the gravity orbit: r' = v v' = -r/|r|^3 To solve it i need to convert to first order. It is clear that right side is the gradient: grad(1/r) = -r/|r|^3 But for the cartesian it looks strange because of |r| and i am…
XMaSt3R
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Find y' and y'' and at x=51 for the following (newton forward interpolation)

Here I'm trying to estimate the value of $y'$ and $y''$. It's related to newtons forward and backward interpolation formula. more info I tried to solve it using newton forward interpolation method but I couldn't get it right. I don't know where I'm…
user_12
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Gauss-Seidel Iterative Method

Can we solve not diagonally dominant matrix in Gauss-Seidel Iterative Method? Note: We cannot make the matrix diagonally dominant.
Apurbo Roy
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Numerical Solution $f(x)=\sqrt{x^2+1}-1$

$$f(x)=\sqrt{x^2+1}-1$$ because the function does not change signs both bisection and REGULA-FALSI can not be used, so I have used Newton Rapshon. If will look at the minimum point we can see that $x=0$, can it be translated to a numerical…
newhere
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Bisection method example

I'm writing a small program to resolve functions using bisection method. I want to test the case when the method finds 2 roots, but I can't find examples. Can anyone give me an example of a function that when resoved using bisection method gives 2…
Ashir
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Prove $I_n(P)=I(P)$ for the integration rule

The given integration rule is $$I_n(f)=c_0f(x_0)+...+c_nf(x_n)\approx I(f)$$ where $x_0,...,x_n$ are n+1 distinct nodes. $P_n(x)$ is a polynomial that interpolates $f(x)$ at the nodes $x_0,...,x_n$. Show that $I_n(P(x))=I(P(x))$ for the $P(x)$ at…
Don
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How can I solve $3x^2-\cos(x)=0$?

In an exercise I have to tell if an equation solution does exist and if it is unique in the interval $[1,2]$. So I calculated the derivate of the function $x^3 - 1 - \sin(x)$, which is $3x^2 - \cos(x)$. Then I realised that $3x^2$ can't be zero…
YuriMath
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verify when $n=1$ and $a=x_0, b=x_1$ then Hermite cubics provide a Hermite interpolation of $[a,b].$

How to verify when $n=1$ and $a=x_0, b=x_1$, then Hermite cubics provide a Hermite interpolation of [,]. I have derived the following $4$ polynomials form a basis for the degree $3$ polynomials on $[\alpha,…
Cherry
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How can I show this? (Use Sherman-Morrison formula to obtain an expression for the inverse of a matrix)

I have made several attempts to prove what is asked in the image below. I need to use the Sherman-Morrison formula to check that the inverse of the H_{k+1} matrix can be obtained by the last formula.
Bruno Pires
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numerical analysis finding $f(x) $of iteration formula using secant method

The iteration formula given below was obtained by applying secant method to some function $f(x)$. What was $f(x)$? What can this formula be used for? $$ x_{n+1} = x_n + \frac{(3-10^{x_n})(x_n - x_{n-1})}{10^{x_n}- 10^{x_{n-1}}} $$
Avinesh
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