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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

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Five Point Centered Approximation and Taylor expansion

I have a homework question that I don't know how to answer. It goes like this: Give complete details on the derivation of the five-point centered approximation to the second derivative of a function $f(x)$ for the example below. Also, give complete…
David_T
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How to show that a solution blows up

I am doing this problem. For part a, I think I can just let $y(t)=at^2+bt+c$, but the problem is, how can I make sure what is the max precision of $y(t)$? I am not sure if it is of degree 2 or 3 or 4, but since I am only asked to find $y(0)=s$, I…
J.doe
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How to use Composite Trapezoidal Rule for this particular problem

The original ODE I had was $4f‴+ff″+2(f′)²=0$ with $f(0)=α,f′(0)=β$ Now I want to use Composite Trapezoidal Rule for this integral : $$ ∫(f-α)(f′)²dη = 1 \qquad \text{from}\,\, 0…
Md. Taj Uddin Raju
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Modified Newton's Method multiplicity

I have an assignment question that I have no idea how to solve. It goes like this: The multiplicity of the zero $x^*$ is the least integer $m$ such that $f^{(k)}(x^*)=0$ for $0 \le k \lt m$, but $f^{(m)}(x^*) \neq 0$. Show analytically that in the…
David_T
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Calculating the elements of Jacobian Matrix

A Jacobian Matrix is defined as : If we have discrete data and we are applying any numerical modelling approach how will end elements of the matrix be calculated. for example : if central difference is applied then df(i)=df(i+1)-df(i-1) .based on…
shrey
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Numerical Analysis Exam

We have an upcoming exam and the question I have attached was from last years exam. But I am really stuck on how to get started with it. I would really appreciate some help with it.
user147825
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Numerical Methods that find roots like Durand Kerner

I need to find n roots for a given polynomial of order n, single root methods like newton and secant wont work in this case, and Durand Kerner doesn't seem to converge nearly as often as I need it to, are there any other stable methods that output…
BinkyNichols
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Find $b_m$ in the Euler Maclaurin summation

Find a formula for $b_m$ by evaluating both sides for $f(x)=e^{\lambda x}$ where $\lambda$ is a parameter. The formula is $\int_0^1 f(x)dx=1/2(f(0)+f(1))+\sum_{m=1}^\inf b_m(f^{(2m-1)}(1)-f^{(2m-1)}(0))$ for some unknown constants $b_m$ independent…
J.doe
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How to measure the error of a partial differential equation in the discrete norm?

If we know the exact solution and approximate solution of a PDE then we can compute the error of both solutions in the discrete norm as given: $\|u-u_N\|^2_M$ = $\frac{2\pi}{M}\sum _{j=0}^{M-1}(u(z_j, t)-u_N(z_j, t))^2$. I am unable to understand…
R. Shah
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Find a formula for the polynomial of degree $2p+1$

For part b, we know the value of $x_0$ and $x_1$, how does that help with $x_j$? Or we don't need to worry about $x_j$ at all? And any thoughts on part c and part d?
J.doe
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the limit of a convergent sequence

I don't get it. How do we exactly know that the sequence is decreasing and $lim a_n=0$?
J.doe
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divergent and bounded sequences

For the sequence $Z_n= ncos(n\pi/4)$. I know $S={-inf, 0, inf}$. I know this sequence has no limit, but can I say $Z_n$ is bounded? Or it is divergent to $-inf$ or $inf$?
J.doe
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Fixed point iteration: finding sufficient conditions

1) Suppose it is known that the fpi $x_n :=g(x_{n-1})$ satisfies the estimate $|x_n-\alpha| \leq c|x_{n-1}-\alpha|^2$ a) Show by induction that $c|x_n-\alpha| \leq (c|x_0-\alpha|)^{2^n} $ and give some condition that is sufficient for the…
user314580
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Fixed point, errors and steep accuracies

$f(x)=e^x-x-2=0$ on $[a,b] = [0,3]$ Fixed point iteration $g(x)= \ln(x+2)$ was derived from it with given $x_0=1.5$ how many iterations until $10^{-100}$ accuracy is reached? I got $(1/2)^{n-1}*|0.247| =< 10^{-100}$ I can't handle this calculation…
user314580
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Error calculation in this Fixed point iteration

How many iterations to get accuracy within $10^{-8}$ $g(x)= 1/3\cdot\ln(2-x^2)$ when $x_0=0.5$ I used $|0.186539-0.5|\cdot{\lambda}^{n}/(1-\lambda) $ where $\lambda = 2/3$ and got $n \geq 46$ but I smell a mistake. Note: the original question is…
user314580
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