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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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Finding Relative Error Of $\cos^2x-\sin^2x$

Find the relative error when storing six digits in the mantissa using rounding of $\cos^2x-\sin^2x$ where $x=0.7854$ So $f(0.7854)=-3.673205105\cdot 10^{-6}$ And $fl(0.7854)=(0.7071)^2-(0.7071)^2=0$ So the relative error is $$\frac{\mid…
gbox
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Formatting Functions To Avoid Loss Of Significant

Rewrite the following to avoid loss of significant $\ln(x+1)-\ln(x)$ where $x>>1$ $\cos^2(x)-\sin^2(x)$ where $x\approx \frac{\pi}{4}$ $\sqrt{x^2+1}-x$ where $x>>1$ $\sqrt{\frac{1+\cos x}{2}}$ Using taylor expansion we get…
gbox
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Summing Number With 6 Significant Digits

The given operation is done with $6$ significant digits (mantissa) and in base $10$ $$10^6+\underbrace{1+1+\ldots+1}_{10^6\text{-times}} $$ Now we had to get $2*10^6$ but we get $10^6$, why is that? I understand that the number is stored…
gbox
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Is adding and subtracting 1 some generic method of removing numerical instability?

Is adding and subtracting 1 some generic method of removing numerical instability? Like in: Unstable: $$\frac{\log(1+3 \cdot 10^{-16})}{3 \cdot 10^{-16}}$$ Stable: $$\frac{\log(1+3 \cdot 10^{-16})}{1+3 \cdot 10^{-16}-1}$$ The correct answer (of the…
mavavilj
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Quoting uncertainty on numerical integration of data with uncertainties

If I have a set of data points ($y_i$, $x_i$) and measurement uncertainties ($\Delta y_i$,$\Delta x_i$) giving error bars for each point, and I want to estimate the integral $$ F = \int_a^b \text{d}x \, y(x) $$ using a simple numerical integration…
RGWinston
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Proof: Newton-Raphsons numerical method is of second order

In the proof of the fact that Newton-Raphson is a second order numerical method they use a Taylor-sequence. Iterative formula for Newton-Raphson: $x_n=x_{n-1} - \frac{F(x_{n-1})}{F'(x_{n-1})}$ Why is there a factor $F''(\xi_{n-1})$ with $\xi_{n-1}$…
WinstonCherf
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Regula falsi method

I have some troubles with the regula falsi method. Given: $F(x)=x^3-x-1=0$ $$ \mathbf x_2=\frac{x_0F(x_1)-x_1F(x_0)}{F(x_1)-F(x_0)} $$ Find the root of the function. Book gives the next options: 1) $F(x_0)\cdot F(x_2)<0$ --> Repeat with $x_0$…
WinstonCherf
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Extrapolating $\pi$ using Taylor expansion

Let $P_n$ be a polygon inscribed in a circle with diameter $1$. Each side of the polygon has length $l_n=\sin(\pi /n)$ and the circumference of $P_n=nl_n$. With $$P_n= \pi - \frac{\pi^3}{3!}\frac{1}{n^2}+ \frac{\pi^5}{5!}\frac{1}{n^4}-…
user463026
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About three-point method

Using the three-point startpoint formula to find $f^{'}(7.4)$ where: $f(7.4)=-68.3193, f(7.6)=-71.6982, f(7.8)=-75.1576$ I got the approximated value: $f^{'}(7.4)=-16.69325$ then the actual error equals: $0.000367$ where $f(x)=ln(x+2)-(x+1)^{2}$ and…
soso sos
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which one has better numerical properties

Given $ w,x,y \in \mathbf{R^{n}}$ which one has better numerical properties? (1) $ w = z - (x \cdot y^{T}) \cdot z$ (2) $ w = z - x \cdot (y^{T} \cdot z)$ First one needs more FLOPS (thus more possible floating point arithmetic errors) and storage…
tomtom
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Significant figures problem

I state a problem from my book : Find the number of significant figures in $V_A=11.2461$ given its absolute error as $0.25×10^{-2}$. Now I thought that significant figures refer to those digits that give meaning to the representation of a number and…
Not Euler
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Successive Approximation

If I'm given the following parameters, Let the function $f(x)$ be four times continuously differentiable and have a simple zero ξ. Successive approximation $x_n, n = 1,2,...$ to ξ are computed from $x_{n+1} =…
StealzHelium
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Numerical methods root finding for function with one variable

Fixed point iteration , bisection, newton , regula falsi, secant all methods are used for only continuous functions with one variable, aren't theY? What can we think for numerically solving of one variable functions with noncontinous ?
HD239
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Fixed point iteration "Simple iteration" - restating a problem

Okay, here's the question: Suppose we want to find a solution to $\frac{1} {2} e^{x} - x = 0 $ on the interval [0,1]. Show how to restate this problem as a fixed point problem. This is the first part of the question, but I think I'll know how to do…
Fred
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How to get the cubic iteration of the Newton method?

I have the taylor expansion: $$f(x+h)=f(x)+hf'(x)+ \frac 1 2 h^{2}f''(x)+O(h^3) $$ and I am trying to get the cubic iteration of the the Taylor method: $ x_{n+1}=x_{n}-f'(x_{n})(1-\frac{\sqrt{(1-2f"(x_{n})f(x_{n})})}{f"(x_{n})} )$ I have tried…
chloe loughan
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