Contents
How do you do a stiffness matrix?
Assembling the Global Stiffness Matrix from the Element Stiffness Matrices
- The term in location ii consists of the sum of the direct stiffnesses of all the elements meeting at node i.
- The term in location ij consists of the sum of the indirect stiffnesses relating to nodes i and j of all the elements joining node i to j.
What does well-conditioned mean?
1 : characterized by proper disposition, morals, or behavior. 2 : having a good physical condition : sound.
What is a well-conditioned equation?
Suppose one or more elements of the matrices A and/or b be changed and let them be A and b . Also, let y be the solution of the new system, i.e. Otherwise, the system of equations is called well-conditioned. If a system is ill-conditioned then the corresponding coefficient matrix is called an ill-conditioned matrix.
Which is an example of an ill conditioned matrix?
Wikipedia, Ill-conditioned Matrices. In some cases, the solution to a system of linear equations Mx = b may be very sensitive to small changes in either the matrix M or the vector b —a relatively change in either can result in a significant change in the solution x .
How to find the condition number of a matrix?
The condition number of a matrix M is found in Matlab with the cond (M) command which uses the 2 norm by default. The condition number using the 1- or &infty-norm may be found using cond (M, 1) or cond (M, Inf), respectively. Matlab also provides a function condest (M) which provides an approximation to the condition number using the 1-norm.
How to determine the behaviour of a well conditioned matrix?
In order to motivate this discussion, we will look at two matrices: the first is well conditioned—small changes in either M or b result in correspondingly small changes in x. Consider this first matrix M: To determine the behaviour of this matrix, we will look at the image of the unit circle, as is shown in Figure 1. Figure 1.
Can a determinant be used to determine conditioning of a matrix?
(What is critical here is that the determinant cannot be used to determine the conditioning of a matrix.) Solving Mx = b means that we want to find that point x which maps to b, as is shown in Figure 2 which shows the pre-image of b = (1, 1) T.