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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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Problems involving the second Taylor Polynomial of $e^x\cos x$

I'm working on what seem to be very easy problems, but my answers aren't matching my textbook's. 1) Find the second Taylor polynomial of $f(x) = e^x\cos x$ about $x_0 = 0$. $P_2(x) = 1+x$. (correct) 2) Use $P_2(.5)$ to approximate $f(.5)$. Find an…
mXdX
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How to apply improved Euler method to a systems of differential equation?

I have the following problem (linearized pendulum problem): $x_1'=x_2$ $x_2'=-\frac{g}{L}*sin(x_1)$ with the following initial condition: $x_1(t_0)=0$ $x_2(t_0)=0$ Improved Euler Method says that: $Y_{k+1}=Y_k+\frac{h}{2}*[f(t_k, Y_k)+f(t_{k+1},…
user2235427
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Coefficients of numerical approximation of differentiation

there is given $y^\prime(\frac{a+b}{2}) = \alpha y(a) + \beta y(b) + \gamma y^{\prime \prime}(a) + \delta y^{\prime \prime}(b) \quad \quad (\dagger)$ we want to find $\alpha, \beta, \gamma, \delta$ such that ($\dagger$) is as accurate as possible…
Sajad
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Order of convergence of a two-step iteration method

Find the order of the iteration is given by $$y_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \quad x_{n+1} = y_{n+1} - \frac{f(y_{n+1})}{f'(x_n)}$$ Assuming $f(r) = 0$, $f'(r) \neq 0$ and the initial guess is close to r. I know the iteration is a third…
user627004
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SSOR with acceleration by CG

I need to implement SSOR method with acceleration by conjugate gradient method. But I don't understand how we can to combine two iteration methods? Both algorithms solving $Ax=b$. In book "Applied Iterative Methods", Hageman, Young described…
Pennywise
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Forward Euler Method: how to derive global error

I was just doing some practice questions for a test, but have been stumped by the following for the past couple of hours. I'm given a system such that: $$\frac{du}{dt} = v ~ ~ \& ~ ~ \frac{dv}{dt} = -f(u)$$ with Hamiltonian $$H = \frac{1}{2}…
user3424575
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Regular simpsons rule in numerical methods

In the following code I have implemented simpsons rule. It is working correctly for my first function but for my second function I am getting an error. So, Is it possible to do this code without using np.linspace? and if not how do I fix it so my…
fr14
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how to solve newton's law of cooling using Numerical Solution

A metal bar is heated 100oc by a heat source. After 40 minutes the heat source is removed when the temperature of the metal bar reached to a plateau. Now the metal bar is placed in a room. The room temperature is 25oc. After 10 minutes the bar…
Cynthia
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In Newton-Raphson method $ g'(x^*) = 0 $ and $ g''(x^*)\neq 0 $ for real roots not repeated

Show that in the Newton-Raphson method $ g'(x^*) = 0 $ and $ g''(x^*)\neq 0 $ for real roots not repeated, where $x^*$ is a fixed point. Deriving the function $ g(x)=x-\dfrac{f(x)}{f'(x)} $, and then evaluating in $ x^* $, I have proved that…
Jacob S.
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How Do I Find the Exact solution after using the Finite Difference Method?

My starting equation is $y'' = \frac{wx}{2EI}(L-x)$ [Beam Formula] I got my approximations, but how do I use that to find the exact equation? I know that y = y(homogeneous) + y(particular). But the homogeneous solution would come from $y'' = 0$.…
Jackie
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Rate of convergence

Suppose that $p_{1},p_{2}>0$ and that $F_{i}(h)=L_{i}+O(h^{p_{i}})$ as $h \to {0^ + }$ for $i=1,2$. What are the rates of convergence of $F_{1}(h)F_{2}(h)$ for various values of $L_{1},L_{2}$?
drawar
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A formula for the term of a sequence

A seemingly simple but a challenging quiz, find the nth term in 1,6,30,120..... I am thinking is a quadratic due to the constant difference but n th term is needed not a recursive one - I got confused
Waitara Mburu
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2nd vs. 3rd Order Gauss Quadrature

I have read that using Gauss Quadrature integration, $$\int_{-1}^{1}f(x)dx=\sum_i f(x_i)w_i$$ for polynomials of degree $\leq2n-1$ (and otherwise it is an approximation). Using weight functions $w_i$ from wikipedia, for 2$^{nd}$ order GQ we…
lamdoug
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Implicit Linear multistep method order?

Considering the following linear multistep method: $y_{k+2} = y_{k+1} + \frac{h}{12} \left( -f(x_{k},y_{k}) + 8f(x_{k+1},y_{k+1})+ 5f(x_{k+2},y_{k+2}) \right)$ What is it's order? What is the maximum order that this method can attain?
Tim Jones
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fourth order accurate approx

Prove that the following is a fourth order accurate approximation of the second derivative of a function $f$: $$f''(x_0) = \frac{-f(x_0+2h)+16f(x_0+h)-30f(x_0)+16(x_0-h)-f(x_0-2h)}{12h^2} + O(h^4)$$ Find the leading order error term. Here are my…
user123
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