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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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Estimating the number of digits accuracy

Suppose that solutions are computed to the linear system Ax = b where the entries of A have about 6 digits of accuracy, b has about 4 digits of accuracy. Given that the condition number of A is about $10^3$ How to estimate the number of digits of…
S. Kaz
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Iterative Methods without Local Contraction

It seems to me that many iterative methods in numerical analysis leverage some sort of local contraction to show that a solution will converge to a fixed point. Is this true of all iterative methods or are there iterative methods which do not…
awright96
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Rewrite ${1 \over {1 + 2x}} - {{1 - x} \over {1+x}}$ to avoid loss of significance

I would like to rewrite the term $${1 \over {1 + 2x}} - {{1 - x} \over {1+x}}$$ in order to avoid a loss of significance for $x \approx 0$, meaning I would like to minimize the numerical error made when computing the fraction. I think it might be…
Julian
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Solve $\frac{df(x)}{dx}=b(x)$ numerically

I want to solve the following simple differential equation numerically on a grid: $$\frac{df(x)}{dx}=b(x)$$ I tried to discretize the LHS using central differences, but the problem with that is, that the resulting linear system of equations $Ax=b$…
dkoerner
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Finding order of convergence of fixed point iteration on Matlab

One simple code to find the order of convergence of a fixed point iteration on Matlab is as following, for each $n=0,1,2,...$ calculate $\log(|p_{n+1}|)/\log(|p_{n}|)$ The calculated value for each $n$ will converge to the order of…
Little Rookie
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How to round up to multiple of 4?

I know how to round up to the number base 10 but I do not understand here, "multiple of 4"? and the Question is "round 0xDEA24A20 up to multiple of 4" it is the base 16 not 10. Could anyone give me some hint?
Q123
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Find the closest numbers in double precision

I am reading a numerical analysis book which talks about nearest numbers. They quickly state that you can easily find nearest machine numbers, but never go on to show how exactly to do so. Can someone please explain how you can find the two closest…
JabberJabber
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consistency of numerical methods for ODE

I was reading the Wikipedia article on numerical methods for ODEs and I saw that when it discusses "consistency and order", the consistency is defined as $$\lim_{h\to 0} \frac{\delta^h_{n+k}}{h} = 0$$ where $\delta^h_{n+k}$ is the local truncation…
nan
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Help to understand result on divided differnece

I am reading following Lemma. I am confused about determining the end points $a$ and $b$ as suggested in the lemma. If possible any numerical examples which uses this lemma will be enough to clarify my doubt. Thank you for your kind help.
Srijan
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How to find a starting point for root finding in complex plane?

The function $f(x) = x^2 +1$ has zeros in the complex plane at $x = +i$ and $x = -i$. Is there a real starting point for complex Newton's method such that the iterates converge to either of these zeros? Is there a complex starting point too? I know…
Shadi Taghian
  • 277
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Interpolating polynomials for $f(x)=\sin(\pi x)$

I am having difficulties with interpolating polynomials to approximate. I am familiar when the problem involves 2 points but this particular problem involves 3 points. Can you explain the process to me? For the given function $f(x)=\sin(\pi x)$ let…
math101
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Newtons Method and Error

Consider a variation of Newton's method in which only one derivative is needed; that is, $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$. Find $C$ and $s$ such that $e_{n+1} = C e^{s}_{n}$. First of all, I thouhght the above iterative process was the usual…
Ozera
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Sum of Lagrange Polynomials

Let $L_i(x)$ be the Lagrange polynomials for pairwise different support abscissas $x_0, \ldots, x_n$. Show that $$\sum_{i=0}^n L_i(0)x_i^j = (-1)^nx_0x_1\cdots x_n \text{ for } j = n+1$$ I have previously shown that $\sum_{i=0}^n L_i(x) = 1$ and…
Walter
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Method of Steepest Descent

I have been given this question from my university to complete and I am having some trouble. The lecturer did not do a very good job of explaining the concept and I could not find any materials online to help so I came here. The question is- What…
John
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Apply Watson's lemma to Gamma function

I was wondering is there anyone know how to show that the gamma function $\Gamma(z)$ satisfies the conditions of Watson's Lemma, where z is on the right half plane. After I changed variable t=xv $$\Gamma(z)=\frac{1}{z}…
ychen-student
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