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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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Estimation of truncation error based on MATLAB for a sum calculation

The sum is $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^2+k}$ Question: Determine this sum with 6 correct decimals. And Estimate the truncation error. Firstly we know the MATLAB uses double precision, from binary number, we could find the smallest…
Xingdong
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Computing $\mathrm{B}_{x,y}(\alpha+1,\beta) / \mathrm{B}_{x,y}(\alpha,\beta)$ numerically

I need to compute numerically ratios of the form: $$\frac{\mathrm{B}_{x,y}(\alpha+1,\beta)}{\mathrm{B}_{x,y}(\alpha,\beta)} \tag{1}$$ where $\mathrm{B}_{z_1,z_2}(\alpha,\beta)$ is the incomplete Beta function: $$\mathrm{B}_{z_1,z_2}(\alpha,\beta) =…
a06e
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Proof using the Contraction Mapping Theorem

How do you identify an interval [f,g] so that the Contraction Mapping Theorem guarantees convergence to the positive fixed point for the following: a) $\frac{14-x^3}{13}\ $ b) $e^{-x}$ I tried drawing a graph and i see it visually but am not so…
cambelot
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Approximating the length of an ellipse given equation

Can anyone help me with this problem in numerical analysis? Determine to within $10^{−6}$ the length of the graph of the ellipse with equation $$4x^2+9y^2=36$$ Thanks a lot.
user115608
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Accuracy of the result question short?

We have 1/(4,5) . When we do the divison,what accuracy does the result have?
fsdd
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Jacobi method for any b

Determine if Jacobi method converges for any b for, $$\begin{bmatrix} 2 & 2\\ 3 & 4 \end{bmatrix}$$ The solution goes on like this... D-(L+U) = $$\begin{bmatrix} 2 & 0\\ 0 & 4 \end{bmatrix} - \begin{bmatrix} 0 & -2\\ -3 &…
user136422
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Least Squares Approximation for odd functions question

Suppose that $f \in C[-1,1]$ is an odd function on $[-1,1]$. Show that polynomial $p_n$ of least squares approximation for $f$ in the norm $|\cdot|$ is an odd function on $[-1,1]$.
cambelot
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floating-point operations do not satisfy the well-known laws for arithmetic operations

Introduction to Numerical Analysis, Stoer, Chapter: Error Analysis, Page 8 if $|y|<\frac{eps}{\beta}|x|$ where $eps = 0.5\times 10^{1-t}$ then $$fl(x+y)=x+^*y=x$$ where $fl(x)=$ normalized floating point number closest to $x$ and…
SKMohammadi
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Absolute value of complex number in Numerical Recipes

In numerical recipes in C, absolute value of complex number $a+ib$ is implemented as $b*\sqrt{1+\left(\frac{a}{b}\right)^2}$ if $|b|$ is greater than $|a|$ and $a*\sqrt{1+\left(\frac{b}{a}\right)^2}$ if $|a|$ is greater than $|b|$. Does the…
Discretizer
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B-Spline Question

Show that the cubic B-Spline with integer knots can be written as $$ s(x) = \frac{1}{6}\left [ x^3 \; x^2 \; x \; 1\right ]\begin{bmatrix} -1 &3 & -3 & 1 \\ 12 &-29 & 12 & 0 \\ -48 & 60 & -12 & 0\\ 64 & -44 & 4 & 0 …
Ozera
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How do I solve an equation involving $e^t$ and $t$ on one side? $18=0.5e^t-0.5t-0.5$

The equation is: $$18=0.5e^t-0.5t-0.5$$ How do I solve for $t$? The answer manual gives $3.706$, but it gives no explanation on how to it got there. This is from a recommended dynamics problem, while preparing for a test which does not allow any…
MikeFoxtrot
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Piecewise Linear Rayleigh-Ritz

I am trying to get this equation $-x^2y'' - 2xy' + 2y = -4x^2$ into Piecewise Linear Rayleigh-Ritz format $-\frac{d}{dx} (p(x) \frac{dy}{dx}) + q(x)y = f(x)$ I pretty much needs to figure out what is $p(x), q(x), f(x)$ Thanks
user1510863
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How does one prove that a linear multistep method of order p can recover all polynomials up to and including order p?

It is intuitive that all polynomials up to and including order $p$ can be fully recovered i.e. without error, but how can one rigorously prove this? In the book by Lambert, there is a similar question- Let $L$ be the linear difference operator…
yaska
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how can i use Numerical Analysis to analyse equation from the fourth degree?

i meant that i know way to analyse equation from first , second ,or third degree , Using a table of $(X)$ and $(y)$ . How can i solve this table with equation from the fourth degree ?
user136916
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Proving the zeros of the chebyshev points

I am trying to prove that the zeros of $T_n(x)$, also called the chebyshev points are, $x_i = \cos ((2i + 1)\frac{π}{2n}) \in (−1, 1), i = 0, 1, . . . , n − 1.$ I believe I have to use the fact that $T_n(x)=\cos(n \arccos(x))$ coupled with the fact…
user134751
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