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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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best approximation polynomial $p_1(x)\in P_1$ for $x^3$

I want to find a best approximation polynomial $p_1(x)\in P_1$ for $f(x)=x^3$ in $[-1,1]$ w.r.t. $||\cdot||_{\infty}$. I want to use Chebyshev polynomial to do that, but I don't know how to hang on.
89085731
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solving singular linear system $Ax=0$

what are computational methods for solving square singular linear system $Ax=0$ for a nonzero $x$ with $A$ of large dimensions?
user133834
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An analytical proof that the sequence from the chord's iterational method is monotonic

Assume $f:[a,b]\to\mathbb R$ is twice differentiable and $f'f''\not=0$ on the interval. Assume $f(a)<0
superAnnoyingUser
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3D surface fitting

I am attempting to find the mathematical representation of a surface given a set of (x,y,z) data points. I recently tried using the method of least-squares which worked well for most of my situations. However, in some instances I am getting the…
Travis
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Proof that Jacobi method will converge to the solution of a system Ax=b

Can anyone show me a statement that this works and a proof? Thanks
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Why does newtons method converge to the root of an equation?

I'm trying to understand why the Newton Raphson method converges to the root of a given equation? Can someone explain it to me theoretically. Thanks
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Numerical computation of power differences: $x^a - y^a$

I want to calculate a power difference, $x^a - y^a$, where $a$ can be large, and the numbers $x,y$ are of similar magnitude. What's a sound numerical way to approach this? Note: $x,y,a$ are all positive real numbers (not integers in general).
a06e
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Newton's method linear convergence proof

How would you show that if f'(a)=0 then the Newton's Method is linear convergent when 1. $f''(a)\neq 0$ 2. $f''(a)=0, f'''(a) \neq 0$? I am having some trouble getting it to the point where you can take the limit of ratio of the error terms and…
cambelot
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Method of False Position (Regular Falsi) - Pros/Cons

Could anyone provide and explain some drawbacks and benefits of the method of false position against say newtons method. I know one of benefits is that it doesn't require the derivative and one of the cons is that one of the interval definitions can…
Harry
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Backward Stability

For $x$ close to $0$ the computation $y=\log(1-x)/x$ is numerically unstable but computing via $y=\log(1-x)/(1-(1-x))$ is not. This question comes from http://www.cs.berkeley.edu/~demmel/cs267/lecture21/lecture21.html where they say that only the…
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Theorem on Iterative Method Convergence

Theorem on Iterative Method Convergence If $\vert \vert I - Q^{-1}A \vert \vert <1$ for some subordinate matrix norm, then the sequence produced by $Qx^{(k)} = (Q-A)x^{(k-1)} + b$ converges to the solution of $Ax = B$ for any intial vector…
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Find the first two iteration of the Jacobi method for the following linear system, using $x^{(0)} = 0$

$$3x_{1} - x_{2} + x_{3} = 1,$$ $$3x_{1} + 6x_{2} + 2x_{3} = 0,$$ $$3x_{1}+3x_{2}+7x_{3} = 4$$ So, from this I got T = \begin{bmatrix} 0 & \frac{-1}{3} & \frac{1}{3} \\[0.3em] \frac{1}{2} & 0 & \frac{1}{3}…
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Evaluate the integral $I=\int_0^{1.6}\frac {1}{1+x^4} dx$

$I=\int_0^{1.6}\frac {1}{1+x^4} dx$ by using generalized trapozoidal rule $n=8$ the final answer don't equal the correct answer .I need the final answer and how can i solve it ? The answer to your question may not always be the one you wanted,…
mahmoud afefey
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Guessing root of polynomials

Given $p(x)=x^5+(1+2i)x^4-(1+3i)x^2+8+44i$ check with the Horner-scheme if $(-2-i)$ is a root of $p(x)$. First I have to guess a root, then proceed with the Horner-method and if i factorized it, i can say if $(-2-i)$ is a root or not, but how can…
OBDA
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Using Newton's Method with a Piecewise Defined Function

Using a Natural Cubic Spline approximation, I've generated an approximation polynomial to six points of data. Using the Cubic Spline approximation polynomial, I now need to use Newton's method to find a root of it (the spline approximation). I'm…
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