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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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Simple Newton's method problem

Estimate the number of iterations of Newton's method needed to find a root of $f(x)=\cos(x)-x$ to within $10^{-100}$. The answer is $7$ iterations, but I have no idea how it was solved by my instructor.
user87274
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Give me a fun problem related to numerical methods.

I hope that this doesnt violate the rules since I need a problem instead of an answer. We have to make our own problem and present it in the class. First course in numerical methods using MatLab. My initial idea: A function gets n random…
user31113
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Improving Newton Iteration

What are the options for a Newton Iteration that starts jumping between two values and never converges? I am projecting 3D points on the spherical UV surface and approaching the poles the issue arise. Is the issue releated to the projection problem…
abenci
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Least square problem and pre-Hilbert, Numerical analysis Homework

how to show the minimizing max has a solution. confusing about how to approve it
John
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Solving system of equations by using simple iteration method

I have a problem: $$\begin{cases} \sin(x) + 2y = 2 \\ \cos(y - 1) + x = 0.7 \end{cases} $$ with margin of error 0.00001 And I need to solve this by using Fixed-point iteration method. Can someone help me with that? Algorithm for full solution might…
domakas
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Newton's Method and Secant Method Convergence proof

Show that if the iterates in Newton's method converge to a point $r$ for which $f'(r)\ne0$, then $f(r)=0$. Also establish the same assertion for the secant method. Hint: For the secant method, the mean-value theorem is useful. This is the case…
Drake
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Finding intervals containing solutions to a equation, without using a graph

How can I find all intervals containing solutions to the equation $4x^2-e^x=0$, without using a graph? I know how to solve this problem, but not without a graph. Thanks.
user85362
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Significant digits question. compute $\sin x$

Why if $x$ is a machine number on a $32$-bit computer that satisfies the inequality $x > \pi 2^{25}$, then $\sin x$ can always be computed with no significant digits? Thank you.
John Lennon
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Which is the correct relative error?

Which is the correct relative error? $$ r_1=\frac{|p_n-p|}{|p_n|} $$ or $$ r_2=\frac{|p-p_n|}{|p|} $$
Dante
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Avoid cancellation errors for approximating the derivative of the sine function

The function $f_1(x_0,h) = \sin(x_0+h) - \sin(x_0)$ can be transformed into another form, $f_2(x_0,h)$, using the trigonometric formula: $$\sin(\phi)-\sin(\psi) = 2\cos\left(\dfrac{\phi +\psi}{2}\right)\sin\left(\dfrac{\phi…
user59036
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Show that $g'(p) \approx ({p_n-p_{n-1}})/({p_{n-1}-p_{n-2}})$

Suppose the sequence {${p_n}$} is generated by the fixed point iteration scheme $p_n = g(p_n-1).$ Further, suppose that the sequence converges linearly to the fixed point $p.$ Show that $g'(p) \approx ({p_n-p_{n-1}})/({p_{n-1}-p_{n-2}})$ Show that…
PooperScooper
  • 671
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Conversion of single precision double into a floating point

How do I convert $0$ $00000001$ $00000000000000000000000$ into a floating point number? Apparently the sign is + I've started with $00000001$ which is 1 in decimal. Then I applied formula $exponent = c -127 \Rightarrow exponent = -126$ and I got…
John Lennon
  • 1,302
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1 answer

Roundoff Error by a machine number

What is the roundoff error when we represent $2^{-1} + 2^{-25}$ by a machine number? (Note: this refers to absolute error, not relative) Please, help
John Lennon
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Modified Newton's method - denominator 0

While I find information on Modified Newton's method, I have read this post from mathstackexchange. In short, a modified Newton's method, $x_{n+1} = x_n - k\frac{f(x_n)}{f'(x_n)} ,$ such that ,in the case of $k>1$ , and $f$ has a root of…
JAEMTO
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Fastest Algorithm to find the root of a second order linear diff eqn.

I have a second order diff eqn that is in the form of \begin{equation} y^{\prime\prime} + \Big(\frac{3}{x} - \frac{3x^{-4}}{2[x^{-3} + B]}\Big)y^{\prime} - \frac{3}{2}\frac{1}{x^5[x^{-3} + B]}y = \mathcal{F}\,. \end{equation} where $B$ and…
seVenVo1d
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