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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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Prove that $g(x)=e^{-x^2}$ has a unique fixed point on the interval [0,1]

Hello I need help with this question, Prove that $g(x)=e^{-x^2}$ has a unique fixed point on the interval [0,1]
PooperScooper
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Newton-Raphson Method

just want somebody to help me verify if I'm doing using Newton-Raphson method correctly by checking the result of an equation. $f(x) = e^{x-1} + x^2 - 7$ I'm trying to find the zero of f(x) with initial guess $x_0 = 1.5$, and by applying the method…
Vol_Smile
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Let $\deg(f)=m$. If $f[x_0,x_1,…,x_n,x]$ is a polynomial of degree $m-(n+1)$. Show that $f[x_0,x_1,…,x_n]$ vanishes for any $x_i$, if $\deg(f)≤n$

Question is : Set $f(x)$ to be a polynomial of degree $m$. Denote $f[x_0,x_1,…,x_n]$ as the Newton Divided Difference. Given that $f[x_0,x_1,…,x_n,x]$ is a polynomial of degree $m-(n+1)$. Use this result to show that $f[x_0,x_1,…,x_n,x]=0$ for any…
user1228735
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Order of a linear multistep method

Given a linear multistep method $\sum_{j=0}^{k}\alpha_jy_{n+j}=h\sum_{j=0}^k\beta_jf_{n+j}$ how to show using Taylor series expansion that the method is of order $p$ if and only if $D_0=D_1=D_2=\cdots=D_p=0$,…
Manjil P. Saikia
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Find the best integer value possible using o- and O-notation for some series.

a. Consider the series \begin{equation} e^{\tan(x)}=1+x+\frac{x^{2}}{2!}+\frac{3x^{3}}{3!}+\frac{9x^{4}}{4!}+\dots\qquad (|x|\leq\pi/2) \end{equation} Retaining three terms in the series, estimate the remaining series using o-notation with the best…
AthenB
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how to find the zero root of only one equation with two arguments by Newton-Raphson's method?

The two dimensional Newton-Raphson method is to find a zero root $(x_0,y_0)$ which satisfy \begin{array}{*{20}{c}} {f\left( {{x_0},{y_0}} \right) = 0} \\ {g\left( {{x_0},{y_0}} \right) = 0} \end{array} by iteration. However, now I have only…
Pengpeng
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Fixed Point Method : Negative Roots

Graphical Representation of Function f(x) and g(x) Red function: $f(x) = x^6-x-1 = 0$, Green function: $g(x) = (1 + x)^{(1/6)}$ So I am doing my Problem Based Learning on this Fixed Point Method for root finding and when using the equations as…
itsAlwaysOk
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Lipschitz Condition for $f(t,y)$

How do you begin to show that $f(t,y)$ satisfies a Lipschitz Condition on $[0,1]$ for $y' = f(t,y) = t\sin(\frac12t\pi)$, $y(0) = 1$? There’s no $y$ in the $f(t,y)$, where do I begin?
C.Ward
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Formula for monthly payment of mortgage

What is the formula for monthly payment of mortgage including Term, Interest Rate, Cost of Home Down, Payment Insurance, Property Tax, HOA Fee. I'm a programmer and want to add this functionality to my site. The following is a basic formula is…
Rohit Vipin Mathews
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Convergence of the Jacobi method

Consider this question The function $u(x)=x(x-1), 0 \leq x \leq 1$, is defined by the equations $u^{\prime \prime}(x)=2,0 \leq x \leq 1$, and $u(0)=u(1)=0$. A difference approximation to the differential equation provides the estimates $u_m \approx…
Maths Wizzard
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Clarification on what the question wants me to do

Consider Let the Gauss-Seidel method be applied to the equations $A \boldsymbol{x}=\boldsymbol{b}$ when $A$ is the nonsymmetric $2 \times 2$ matrix $$ A=\left[\begin{array}{cc} 10 & -3 \\ 3 & 1 \end{array}\right] . $$ Find the spectral radius of…
Maths Wizzard
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Iterative function with three terms - can subscripts be negative?

I am currently working on Newton's method for approximating roots to functions, and part of the chapter discusses other iterative functions. The basic definition is: Consider a function $F$ and an initial number $x_0$. Define the subsequent numbers…
brocolliSally
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Applying Euler's Formula to a Boundary Condition PDE

I need to apply Euler's formula to the following, 1st order PDEs which are boundary conditions for a numerical physics problem in magneto-statics: $$\partial_r{A_\phi}^n = \partial_r{A_\phi}^m$$ $$[\partial_z{A_\phi}^n =…
Sophia
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Numerical solution of a simple transport equation

I wish to solve the equation: $$\frac{\partial v}{\partial t}=-\frac{\partial v}{\partial x}$$ along with the boundary conditions: $$\frac{\partial v}{\partial x}\Bigg|_{x=0}=\frac{\partial v}{\partial x}\Bigg|_{x=1}$$ I want to use an $O(\delta…
Matthew Hunt
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What quadrature rule type is being used here? How do I define the weights?

I have been given a quadrature rule applied to $I=\int_{a}^{b}f(x)\,dx$ which, for $a = -1, b = 1$, has nodes $t_q= -1 + \frac{2q-1}{m}$ for $q=1,2,...,m$ In the case of 5 points, the nodes would be -0.8, -0.4, 0, 0.4, 0.8 for $t_1$, $t_2$,…
nurip
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