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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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Trying to solve $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$ using Newton-Raphson method

This is what I've atempted so far in solving $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$. The following are the steps: step 1: $f(\lambda) = \lambda^3 - 3.250\lambda^2 + \lambda - 0.063 $ step 2a: $f(0) = -0.063$ step 2b: $f(1) = -1.313$ step…
alok
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Why the negative gradient gives the direction of the steepest decrease in the gradient descent algorithm?

I understand that the gradient vector gives the direction of the maximum growth. What I don't get is why going the exact opposite direction is going to get the maximum decrease? By sure that holds for a single variable function because it only have…
Ariel Scarpinelli
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Why it is more accurate to evaluate $x^2-y^2$ as $(x+y)(x-y)$ in floating point system?

The expression $x^2-y^2$ exhibits catastrophic cancellation if $|x|\approx|y|$. Why it is more accurate to evaluate as $(x+y)(x-y)$ in floating point system (like IEEE 754)? I see this is intuitively true. Any one can help demonstrate an example…
Tony
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How to correctly apply Newton-Raphson method to Backward Euler method?

I'm solving a system of stiff ODEs, at first I wanted to implement BDF, but it seem to be a quite complicated method, so I decided to start with Backward Euler method. Basically it says that you can solve an ODE: $y'=f(t,y)$, with initial guess…
Andrew
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how to solve $aX+bX^2=e^{cX}$

I build a model for our problem, but i cannot get a result from my model. Could anyone give me some idea to solve this formula: $aX+bX^2=e^{cX}$ Thx in advance!
user520813
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How to safely solve a pair of elementary equations in a floating point computing system?

I wrote a simple short computer program to solve a pair of equations of the format , $y = a1 * x + b1$ $y = a2 * x + b2$ . But , it outputs clearly wrong answers sometimes when $abs(a1)$ or $abs(a2)$ is very small ,but not zero. I analyze the…
seven_swodniw
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How can we show that $x_{n+1}=f(x_{n})$ will converge if $|f'(x)|\leq\lambda<1$ on the interval $[x_{0}-\rho, x_{0}+\rho]$?

It is an exercise from Kincaid and Cheneys's book. How can we show that $x_{n+1}=f(x_{n})$ will converge if $|f'(x)|\leq\lambda<1$ on the interval $I=[x_{0}-\rho, x_{0}+\rho]$ where $\rho = \frac{|f(x_{0})-x_{0}|}{1-\lambda}$? My idea is to show…
Aranel
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Can someone explain Newton's method to me?

I was wondering whether anyone can explain in a simple manner the math I'm being asked to do below. PROBLEM STATEMENT: We require a program that will show the way that Newton’s Method is dependent on the initial guess chosen. The functions…
Cale
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Solve expression without loss of precision

Given the…
Drake Jacovian
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Solving a Nonlinear System

$$\begin{align} (1/300)a + (-1/200)b &= 5\\ (-1/300)a + ((-1/300) + (1/200))b + (-1/200)c &= -e^b\\ (-1/200)b + (1/200)c &= -e^c \end{align} $$ how do I solve for $a, b$ and $c$? Thanks! I know if I derive an equation that isolates a variable,…
user434565
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Approximating a big number

Is there some relatively nice way to show that $100^{100}-99^{99}>9.92\cdot 10^{199}$ by using only pencil and paper?
selfstudying
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Find the error approximation for the function $f(x) = \dfrac{1}{1-x}$

Problem Let $f(x) = \dfrac{1}{1-x}$, find the Taylor polynomial $P_n(x)$ about $x_0 = 0$. Find a value of $n$ such that the approximation is within $10^{-6}$ on $[0, 0.5]$. To find $P_n(x)$ is straightforward, $$f^{(1)}(x) = \dfrac{-1}{(1 -…
roxrook
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Failed attempt to handle subtractive cancellation

As both a programmer and a math student, I am trying to come up with a fool-proof way to handle errors from subtractive cancellation caused by trying to evaluate $x-y$, where x,y are extended (long double) precision floating-point numbers.…
Mike Warren
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Numerical evaluation of an integral "similar" to an exponential integral

What is an efficient and stable numerical algorithm to evaluate the integral: $\int_0^L e^{-\alpha x}\frac{e^{\frac{i\beta}{(x+x_0)}}}{(x+x_0)}\mbox{d}x$ with $i$ the imaginary unit, $\;(\alpha,L)>0$, $\;(\beta,x_0) \in \mathbb{R}$? Some notes. If…
user2696676
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computing the degree of exactness for a quadrature formula

I am reading Numerical Analysis and I have some problems on how to use in practice the definition of the '$\textbf{Degree of Exactness}$'. $\textbf{Definition:}$ Let $I_n(f)$ be a quadrature formula for integrating $f$ in the interval $[a,b]$ and…
Albanian_EAGLE
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