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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's method for roots of multiplicity

Any suggestion? I have no idea what the question is asking. Thanks in advance.
Bob
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Jacobi Method with error

Preparing for my Numerical Analysis exam, If the Jacobi's method is used to solve the linear system, $Ax=b$, where $$A=\begin{pmatrix}5 & -2 & 3\\-3 & 9 & 1\\2&-1&-7\end{pmatrix}$$ will the method be convergent? This part I think I can do. Since…
Neurax
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Convergence of QZ algorithm

When performing QZ iterations on matrices A and B where A is upper hessenberg to begin with and B is upper triangular, how can one tell when the algorithm has converged? I understand that if it consists of only real eigenvalues then the subdiagonal…
Travis
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Which fixed point iteration will converge? and why?

Which of the following fixed point iterations will converge, and why? Also if possible please give the rate of convergence. (a) $x_{n+1}=\cos(x_n)$ (b) $x_{n+1}=\sin(x_n)$ (c) $x_{n+1}=\tan(x_n)$ Thank you!
BuddyD
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For what range of values will the fixed point iteration converge?

For what range of values of $c$ will the fixed point iteration $x_{n+1} = x_n + c{x_n}^2 - 9$ converge, and for what particular value of $c$ will it converge much faster?
BuddyD
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What is the $\lim_{n\rightarrow \infty}x_n$? (Secant Method)

If $x_{n+1}=x_{n} + \dfrac{(2-e^{x_{n}})(x_{n} - x_{n-1})}{e^{x_{n}} - e^{x_{n-1}}}$ with $x_o=0$ and $x_1 = 1$. What is the $\lim_{n\to \infty}x_n$? By doing a little manipulating from the equation above I found that $f(x)=e^{x}-2$ so can I say…
Gamecocks99
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How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system?

How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system…
OBDA
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Newton's Function Confusion

"Suppose that $r$ is a double root of $f(x) = 0$, that is $f(x)=f'(x)=0$, $f''(x) \neq 0$, and suppose that $f$ and all derivatives up to and including the second are continuous in some neighborhood of $r$. Show that $\epsilon_{n+1} \approx…
Ozera
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Approximating the value of an $n$th degree taylor polynomial

First I need to derive the Taylor polynomial for degree n for the two functions below: f(x) = $\sqrt(1+x)$ and f(x) = cos(x) Afterwards I need to find the approximate value of both functions at x = $\pi$/4 by hand calculator (up to two decimal…
InfoSecNoob
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Cubic spline that interpolates $f$

Given the following cubic spline that interpolates $f$: $$\left\{\begin{matrix} (x+3)^3-9(x+3)^2+22(x+3)-10 & ,-3 \leq x < -1\\ (x+1)^3-3(x+1)^2-2(x+1)+6 & , -1\leq x <0\\ ax^3+bx^2+cx+d & ,0\leq x <2\\ (x-2)^3+6(x-2)^2+7(x-2) & ,2 \leq x \leq…
Mary Star
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Expressing a polynomial in terms of Chebyshev series

How do I express an nth order polynomial in terms of the Chebyshev terms of the first kind? In other words, how do I express f(x) = $a_0$+$a_1$x+...+$a_nx^n$ in terms of $b_0T_0+b_1T_1+...+b_nT_n$?
user10311389
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Showing that Newton's method converges

I'm studying newton's method for the equation $\cos(x) = x$ over $[0, \pi/2]$ and I'm asked to support the argument that given the values $x_0 = 0.78539816$, $x_1 = 0.73953613$, $x_2 = 0.73908518$ further iterations of the method will be convergent…
jtht
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Relative error machine numbers

Suppose we are working with a machine that does arithmetical calculations with a relative accuracy of $\xi, |\xi| \leq \xi '$. We want to calculate $a^2 - b^2$ in the following two ways; $(A);$ $a^2 - b^2 = (a - b)(a+b)$ and $(B)$; $a^2 - b^2 = a…
user119470
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Prove Newton's method converges...

How would you prove that Newton's Method applied to $f(x) = ax + b$ converges in one step? Would it be because the derivative of $f(x)$ is simply $a$?
Jesus
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show that $B^{T}AB$ is positive-definite if and only if $B$ is invertible.

Given that $A \in R^{n,n}$ is symmetric and positive-definite and $B \in R^{n,n}$,show that $B^{T}AB$ is positive-definite if and only if $B$ is invertible. That's what I have done so far: We suppose that $C=B^{T}AB$ is positive-definite,that means…
evinda
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