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.

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How to choose the penalty coefficient when solving constrained optimization problem?

I was wondering why don't we select a very large number to a penalty function $c$ of the augmented function $f(x) + cP(x)$ instead of doing algorithms to increase c slowly? I know that as $c$ is large ...
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Is there a formula for this interest over a period

Say I have £100 overdraft and my balance is currently £0. when I go into my overdraft I am charged 1.25 % at the end of each day, but I have to divide the rest of the money equally for the next 10 ...
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Condition number of a matrix with factorization $A=QR$

Let $A=QR$ when $Q$ is orthogonal and R is triangular superior proof $\dfrac{1}{n}k_1(A)\leq k_1(R) \leq n k_1(A)$ and $k_2(A)=k_2(R)$ So we have for the second part $k_2(A)=||A||_2|||A^{-1}||_2$ and ...
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Example of matrices where computing the inverse is the most efficient method

I know there exists matrices where for example LU-factorization is not the most efficient way of solving the linear system of equations: $$Ax=b \tag{1}$$ Examples of such matrices are triangular or ...
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rounding error bound and truncation error bound in forward Euler method

How should I derive the truncation error bound and rounding error bound in the forward Euler method? $f^{\prime}(x) \approx \frac{f(x+h)-f(x)}{h}$ I know that the bound for truncation error is 2M/h ...
49 views

Approximate solution of a non-linear system

Is there a method to find an approximate solution of the following system of nonlinear equations, with this type of exponents? \begin{equation} \left\{\begin{matrix} \dfrac{x^{20}-1}{x-1}+\dfrac{z^{...
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A bound for the error $|(x-1)\ln x-P_3 (x)|$ in using $P_3(x)$ to approximate $(x-1)\ln x$ on the interval $[0.5, 1.5]$

I was wondering if someone could tell me how we can find a bound for the error $|(x-1)\ln x-P_3 (x)|$ in using $P_3(x)$ to approximate $(x-1)\ln x$ on the interval $[0.5, 1.5]$. I computed $P_3 (x)$ ...
48 views

Newton's method in higher dimensions

To calculate the inverse of a quadratic matrix A, we could solve the equation $F(X):=X^{-1}-A=0$. I need to show that if X is invertable, then $DF(X)(\Delta X)=-X^{-1}\Delta XX^{-1}$ where DF(X) is ...
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Derivative and Integral of polynomial with degree n

This question is related to Derivation and Integration in polynomial spaces, but since the questions were left unanswered, I decided to break those down to only one question. Question 14.2 We have a ...
36 views

Dirichlet problem for Laplace equation in triangular grid

I need to find the solution in rectangle on an unstructured triangular grid, the example of grid here. I should estimate the values of function for the vertices of triangles. Five point discretization ...
38 views

Power method for complex Hermitian matrices

How should the power iteration be modified to handle complex yet Hermitian matrices? Because the matrix is Hermitian, the eigenvalues are real. I realize that the power method will fail if the ...
$I=\int_{-\infty}^{\infty}c\exp(-i\omega t)dt.$ The problem is given above. How to solve it numerically? The analytically solved answer is a real number. Thanks in advance.