asigurato7

2022-07-22

The linear system ${y}^{\prime}=Ay,y(0)={y}_{0}$, where $A$ is a symmetric matrix, is solved by the Euler method.

Let ${e}_{n}={y}_{n}-y(nh)$

Where ${y}_{n}$ denotes the Euler approximation and $y(nh)$ the exact solution ($h$ is Euler step size).

Prove that $||{e}_{n}|{|}_{2}=||{y}_{0}|{|}_{2}max|(1+h\lambda {)}^{n}-{e}^{nh\lambda}|$

Where $\lambda \in \sigma (A)$ where $\sigma (A)$ is the set of eigenvalues of $A$.

I have tried various approaches such as writing ${e}_{n}$ as the error bound of the Euler method and taking the norm, but I can't seem to get $||{y}_{0}|{|}_{2}$ in my answers.

Let ${e}_{n}={y}_{n}-y(nh)$

Where ${y}_{n}$ denotes the Euler approximation and $y(nh)$ the exact solution ($h$ is Euler step size).

Prove that $||{e}_{n}|{|}_{2}=||{y}_{0}|{|}_{2}max|(1+h\lambda {)}^{n}-{e}^{nh\lambda}|$

Where $\lambda \in \sigma (A)$ where $\sigma (A)$ is the set of eigenvalues of $A$.

I have tried various approaches such as writing ${e}_{n}$ as the error bound of the Euler method and taking the norm, but I can't seem to get $||{y}_{0}|{|}_{2}$ in my answers.

akademiks1989rz

Beginner2022-07-23Added 16 answers

Given that A is symmetric it can be diagonalised. Therefore we can write down the solution of the system of ODEs directly, and simply subtract the discreteized solution front the exact.

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?