Condition Number and Significant Digits

In summary, a larger condition number (k(A)) for an nxn matrix indicates that the solution will have more significant digits, but these digits may not be reliable. This means that the statement "a large k(A) means the solution has more significant digits than if k(A) were small" is true.
  • #1
DODGEVIPER13
672
0

Homework Statement


IF A is nxn matrix, and k(A)=||A||||A^-1|| then a large k(A) means the solution has more significant digits than is k(A) were small true or false?


Homework Equations





The Attempt at a Solution


Well k(A) is defined as 10^(k) right so the larger the condition number the more significant digits that you shouldn't trust. So if I had 10^(16) I would have approximatley 16 sig figs that I shouldn't use. Therefore I would think the answer is true if condition number is large then there will be more significant digits just not usable ones right?
 
Physics news on Phys.org
  • #2
I get that the small the k(A) the more reliable a solution is meaning it would have to have an a good amount of sig figs such that it is accurate so maybe it is false hmmm I don't know?
 
  • #4
Ok, I will give that a look.
 
  • #5
So it is saying that the higher the condition number the larger the error which is what I thought. So I guess what I am supposed to take from this is that the smaller the condition number the more precise the solution, however this problem specifies significant digits not precision so I am going to go with True.
 

1. What is a condition number and why is it important?

A condition number is a measure of how sensitive a mathematical problem is to small changes in the input data. It is important because it can indicate whether a problem is well-posed (stable) or ill-posed (unstable). A high condition number can also lead to inaccuracies in the solution.

2. How is the condition number calculated?

The condition number is typically calculated as the ratio of the largest singular value to the smallest singular value of a matrix. It can also be calculated using other methods, such as the maximum absolute row sum or the maximum absolute column sum.

3. What is the relationship between condition number and significant digits?

The condition number is closely related to the number of significant digits in a problem. A high condition number means that the problem is highly sensitive to small changes, which can lead to loss of significant digits in the solution. In general, a higher condition number requires more significant digits to accurately represent the solution.

4. How can a high condition number be avoided?

A high condition number can be avoided by using a more stable algorithm or by using a different representation of the problem. For example, using a different coordinate system or scaling the input data can sometimes improve the condition number.

5. How does the condition number affect the accuracy of a solution?

The condition number can greatly affect the accuracy of a solution. A high condition number can lead to loss of significant digits and therefore a less accurate solution. It is important to keep the condition number as low as possible in order to obtain the most accurate solution.

Similar threads

  • Introductory Physics Homework Help
Replies
11
Views
758
  • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
  • Classical Physics
Replies
20
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
584
Replies
12
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
4
Views
2K
  • General Math
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
Back
Top