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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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newton: convergence when calculating $x^2-2=0$

Find $x$ for which $x^2-2=0$ using the newton algorithm and $x_0 = 1.4$. Then you get $x_{k+1} = x_k + \frac{1}{x_k} - \frac{x_k}{2}$. How to show that you need 100 steps for 100 digits precision? So I need to show for which $N$ it is…
fritz
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Condition number of $a^2-b^2$

Can someone tell me how to count Condition number of $a^2-b^2$ or recommend a site where I can read about this. I know how to count Condition number of a matrix, but here I'm confused
SugerBoy
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how to solve $x''=f(x)$ numerically?

I know how to solve when $f(x)$ is linear, but how to solve the equation $x''=f(x)$ with $f$ a unknown nonlinear function using numerical methods? Any references would be helpful. Thanks! Update: Sorry that I am not clear about it, what I mean is…
Sherry
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Interpolation and Approximation

A quadratic polynomial $p(x)$ is constructed by interpolating the data points $(0,1)$, $(1,e)$, $(2,e^2)$. If $\sqrt{e}$ is approximated by using $p(x)$ then its approximated value is.
A.MAHATA
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What is the intuitive meaning of 'order of accuracy' and 'order of approximation' with respect to a numerical method?

I have been studying numerical methods in order to get a better understanding of CFD and the algorithms used in CFD codes. Where I am stuck is I don't understand what is meant by 'order-of-accuracy','order-of-approximation' and 'order-of-error'?. I…
GRANZER
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Reasons for testing numerical results against analytical results?

I made a code using python language to solve partial differential equations My supervisor told me to check the results from the numerical model against the analytical results He didnt tell me the reason and said that I have to figure it out by…
CDplayer
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Proving that the Jacobi and Gauss-Seidel converges in a $2\times2$ matrix

Prove that for the system \begin{cases} ax+by=p\\ cx+dy=q \end{cases} a necessary and sufficient condition of convergence for the Jacobi and Gauss-Seidel iterative methods is $|bc|<|ad|$. I have tried to prove it using that the iterative method…
John Keeper
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Trapezoidal method, deriving error boundary.

The trapezoid rule on an interval $[a,b]$ is $$\frac{b-a}{2}(f(a)+f(b))=\int_a^bf(x) \ dx +\frac{f''(\xi)}{12}(b-a)^3, \quad \xi\in[a,b]. \tag i$$ The trapezoid formula is…
Parseval
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Compute $\int_0^1\frac{1}{1+x^2} \ dx$ using Romberg integration.

Compute $$\int_0^1\frac{1}{1+x^2} \ dx$$ using Romberg integration and the trapezoidal formula with stepsize $h=0.125$. I don't really understand because I can either use the trapezoidal formula with the given stepsize or I can start computing…
Parseval
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Can I use the Polynomial Interpolation Error Theorem to conclude that $p_n(x)$ tends to $f(x)$ as $n$ goes to $\infty$?

I was doing some revision questions on the polynomial interpolation error theorem, $$f(x)-p_n(x) = \frac{f^{n+1}\pi_{n+1}(x)}{(n+1)!}$$ After doing out the proof of this theorem, I'm asked if you can use this theorem to conclude that $p_n(x)$ tends…
userpisquared
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Closed loop iterations between 2 points in the Newton-Rapson method

I've been investigating into situations where the N-R method iterates between two $x$ values endlessly. So far I have derived that the relationship between the two values of $x$ should be as follows: $$\frac{f(x_{even})}{f'(x_{even})} =…
Chris Wilson
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Higher order iterative methods

I have the following equation (from Taylor's expansion) $$ f(x_n)+f'(x_n)(x-x_n)+\frac{1}{2}f''(x_n)(x-x_n)^2=0$$ The task: Derive an algorithm for finding the approximation $x_{n+1}$ (determine which root from the quadratic equation must be…
pls_halp
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The equation has exactly two positive roots: How do we get a contradiction?

I want to prove that the equation $x^2+10\cos x=0$ has exactly two positive roots. I also want to calculate approximations for each root using the bisection method. $$$$ I have done the following: $f(x)=0 \Rightarrow x^2+10\cos x=0 \Rightarrow…
Mary Star
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How i get the $x$ of the equation

I have this: $\frac{32}{20 - \frac{16}{5-\frac{5}{x}}}$ = 1 My main problem is that I want to multiply by something on both sides. I will move $(5-\frac{5}{x})$ to side of 1, then: $\frac{32}{20-16} = 1(5-\frac{5}{x}$) $8 = 5 -\frac{5}{x}$ , now…
ESCM
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approximating second derivative from Taylor's theorem

I have been having trouble coming up with an approximation formula for numerical differentiation (2nd derivative) of a function based on the truncation of its Taylor Series. I am not sure if the error is an algebraic one or otherwise. I start of…
apostato34
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