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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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Need help starting a Numerical Differentiation problem

Derive the following difference approximation for the first derivative: $f'(x_0) = (f'(x_0 + 2h) - f(x_0 - h))/3h$ I really just need some pointers in how to start this out. If I were to guess, it looks like it starts out with the Lagrange form of…
Hoser
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If error tolerance is $ 10^{-n} $ , find the maximum allowable angle $ \ \theta \ $ in degrees

We know that $ \ \ \sin \theta \approx \theta \ \ $ if the angle be small. If error tolerance is $ 10^{-n} $ , find the maximum allowable angle $ \ \theta \ $ in degrees. $$ $$ To get this , we need to expand $ \ \sin \theta=…
MAS
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Preconditioning matrix

I want to solve $Ax=b$ system, but the conditioning of $A$ is bad. After some research, I found that the conditioning of $A$ can be improved by multiplying the system by a preconditioning matrix $P$ such as : $Ax = b \Rightarrow P^{-1}Ax =…
Bek
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Is $\sum_{i=1}^n [-1-x_i e^{-\alpha}]=0$ solvable analytically or numerically for $\alpha$?

Is $\sum_{i=1}^n [-1-x_i e^{-\alpha}]=0$ solvable analytically or numerically for $\alpha$? My own interpretation is that if one tries to solve the above by taking the logarithm, then one is not able to get rid of the logarithm, but rather it has…
mavavilj
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Confusion about fixed point theorem

My lecture notes state the following. definition We say that $x$ in $X$ is a fixed point of a function $f$ from $X$ taking values in $X$ if $f(x)=x$. theorem If $f:[a,b]\rightarrow :[a,b]$ is continuously differentiable, then $f$ has at least one…
Meneer-Beer
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Is it possible to find out the value of given variable in the following question?

It is given that $u_1 = 1, u_2+u_3 = 6$ and $u_4+u_5+u_6+u_7+u_8 = 30$. What is the approximate value of $u_4$ when computed by using Lagrange's interpolation formula? 1) $7$ 2) $6$ 3) $5$ 4) $4$ I have no idea how to start with?
kapil
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approximating second order derivative using two stepsizes $h_1$ and $h_2$

I am asked to: Given stepsizes $h_1$ and $h_2$, develop a numerical scheme to approximate $f^{\prime\prime}(x_0)$ with function values $f(x_0)$, $f(x_0 + h_1)$ and $f(x_0 + h_2)$. Under what conditions will your method not work? This is a practice…
Monica
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Find a bound for the absolute error on the interval

Find a bound for the absolute error on the interval $[x_{0}, x_{n}]$ $f(x) = cos(x)+sin(x)$ for $x_{0}=0, x_{1} = .25, x_{2} = .5, x_{3} = 1, n = 3$ So using the Lagrange error bound formula: $ \frac{cos(\xi) + sin(\xi)}{4!}…
Huy Vo
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Error in quadratic interpolation to $f(x)=1/x$?

This is the question: What is the error in quadratic interpolation to $f(x)=1/x$, using equally spaced nodes on the interval $[1/2,1]$? I used this $|f(x)-p_2(x)|\le1/(9\sqrt{3} )h^3 \max|f'''(t)|$ where $x_0 \le t\le x_2$ so, $f'''(x)=…
Alti
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show that the polynomial $f(x) = -x^5 + 2x +3$ has a unique simple root in the interval $[0,2]$

I know that $f(0)*f(2) < 0$ so there must be a root between $0$ and $1$ by the intermediate value theorem. But how can I establish that it is unique and simple? All I really know from the IVT is that the root must have an odd multiplicity, since…
Nażysław Zbyłutowicz
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Numerical Analysis Solving Systems

I have this question for a Numerical Analysis class. However, we've just finished chapter four in class, and this question relates to the end of chapter 5. I'm not quite sure why it's on this assignment. If someone could point me in the right…
Devon Puchailo
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Newton's Forward Difference Question

A fourth degree polynomial p(x) satisfies $$Δ4P(0) = 24$$ $$Δ3P(0) = 6$$ $$Δ2P(0) =0$$ where $$ΔP(x) = P(x+1) - P(x)$$ Compute $$Δ^2P(10)$$ The answer is apparently 1140, but I do not know how to arrive at that answer. The furthest I got to was…
Belphegor
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I need help with using composite simpsons rule to solve an improper integral

I need help using Composite Simpson's rule to solve this improper integral: $$\displaystyle \int_0^2 \dfrac{xe^x}{(x-1)^{2/3}}\,dx$$ We have to use $n=2$ for this question.
PinkT
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The first approximate solution?!

In the fixed point iteration method and Newton's method, how could I choose the first approximate solution $P_0$?
soso sos
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Matlab program for linear FEM in 2D

I have to program the linear finite element method in 2D, and I d like to ask some question about the method. The PDE I have to solve is: $-\bigtriangleup u+cu=f$ and $u(x,y)=g(x,y)$ in the boundary. The set is the unit square and the boundary.…
energy
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