How is the Steepest Descent Formula Used to Solve Linear Systems?

In summary, "steep descent" is an optimization method used in linear systems to find the minimum value of a function by iteratively moving along the steepest descent direction. It differs from other optimization methods by only requiring the knowledge of the direction of steepest descent. The advantages of using steep descent for linear systems include its speed, efficiency, and ease of implementation. However, it may not work well for non-linear systems and can get stuck in local minima. The appropriate step size for steep descent can be determined through line search methods or by using a small fixed step size.
  • #1
Amer
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Steepest descent for linear system

what is the formula of steep descent to solve linear system
can you give me a link
 
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  • #2
Re: Steepest descent for linear system

Amer said:
what is the formula of steep descent to solve linear system
can you give me a link

If Your goal is to find the minimum of an $f(x_{1},x_{2},...,x_{n})$ and the $f(*)$ is linear in the $x_{i}$, i.e. it can be written as...

$\displaystyle f(x_{1},x_{2},...,x_{n})= \sum_{i=1}^{n} a_{i}\ x_{i}$ (1)

... where all the $a_{i}$ are constant and You don't have any constrain, then the goal cannot be met because the (1) has no minimum...

Kind regards

$\chi$ $\sigma$
 
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  • #3
or explain it to me

The formula for steepest descent for a linear system is:

x(k+1) = x(k) - alpha * A^T * (Ax(k) - b)

where x(k) is the current solution, alpha is the step size, A is the coefficient matrix, and b is the vector of constants. This formula is used to iteratively improve the solution until the desired accuracy is reached.

As for a link or further explanation, I recommend checking out some online resources or textbooks on numerical methods for linear systems. Some good sources include Khan Academy, MIT OpenCourseWare, and Numerical Recipes. Hope this helps!
 

FAQ: How is the Steepest Descent Formula Used to Solve Linear Systems?

What is "steep descent" in the context of a linear system?

"Steep descent" refers to a type of optimization method used in systems with linear equations. It involves finding the minimum value of a function by starting at a point and iteratively moving towards the minimum along the steepest descent direction.

How does steep descent differ from other optimization methods?

Unlike other optimization methods that rely on calculating gradients or Hessians, steep descent only requires the knowledge of the direction of steepest descent. This makes it simpler and more efficient for solving linear systems.

What are the advantages of using steep descent for linear system?

Steep descent is a fast and efficient method for solving linear systems, especially in high-dimensional spaces. It also has the advantage of being easy to implement and does not require extensive mathematical calculations.

Are there any limitations to using steep descent for linear system?

Yes, steep descent may not work well for non-linear systems or if the starting point is far from the minimum. It can also get stuck in local minima and may require a good initial guess to converge to the global minimum.

How can one determine the appropriate step size for steep descent in a linear system?

The optimal step size for steep descent can be found by using line search methods, where the step size is adjusted based on the slope of the function. Another approach is to use a fixed step size that is small enough to ensure convergence but large enough to avoid getting stuck in local minima.

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