MATLAB Mathematics - SERC - Index of
Share Embed Flag
Download ePaper

MATLAB Mathematics - SERC - Index of

MATLAB Mathematics - SERC - Index of MATLAB Mathematics - SERC - Index of

serc.iisc.ernet.in
from serc.iisc.ernet.in More from this publisher
15.11.2014 Views

1 Matrices and Linear Algebra If A and B are square and the same size, then X = A\B is also that size: B = magic(3); X = A\B X = 19 -3 -1 -17 4 13 6 0 -6 It can be confirmed that A*X is exactly equal to B. Both of these examples have exact, integer solutions. This is because the coefficient matrix was chosen to be pascal(3), which has a determinant equal to one. A later section considers the effects of roundoff error inherent in more realistic computations. Singular Coefficient Matrix A square matrix A is singular if it does not have linearly independent columns. If A is singular, the solution to AX = B either does not exist, or is not unique. The backslash operator, A\B, issues a warning if A is nearly singular and raises an error condition if it detects exact singularity. If A is singular and AX = b has a solution, you can find a particular solution that is not unique, by typing P = pinv(A)*b P is a pseudoinverse of A. If AX = b does not have an exact solution, pinv(A) returns a least-squares solution. For example, A = [ 1 3 7 -1 4 4 1 10 18 ] is singular, as you can verify by typing det(A) ans = 0 1-16

Solving Linear Systems of Equations Note For information about using pinv to solve systems with rectangular coefficient matrices, see “Pseudoinverses” on page 1-23. Exact Solutions. For b =[5;2;12], the equation AX = b has an exact solution, given by pinv(A)*b ans = 0.3850 -0.1103 0.7066 You can verify that pinv(A)*b is an exact solution by typing A*pinv(A)*b ans = 5.0000 2.0000 12.0000 Least Squares Solutions. On the other hand, if b = [3;6;0], then AX = b does not have an exact solution. In this case, pinv(A)*b returns a least squares solution. If you type A*pinv(A)*b ans = -1.0000 4.0000 2.0000 you do not get back the original vector b. 1-17

Solving Linear Systems <strong>of</strong> Equations<br />

Note For information about using pinv to solve systems with rectangular<br />

coefficient matrices, see “Pseudoinverses” on page 1-23.<br />

Exact Solutions. For b =[5;2;12], the equation AX = b has an exact solution,<br />

given by<br />

pinv(A)*b<br />

ans =<br />

0.3850<br />

-0.1103<br />

0.7066<br />

You can verify that pinv(A)*b is an exact solution by typing<br />

A*pinv(A)*b<br />

ans =<br />

5.0000<br />

2.0000<br />

12.0000<br />

Least Squares Solutions. On the other hand, if b = [3;6;0], then AX = b does not<br />

have an exact solution. In this case, pinv(A)*b returns a least squares solution.<br />

If you type<br />

A*pinv(A)*b<br />

ans =<br />

-1.0000<br />

4.0000<br />

2.0000<br />

you do not get back the original vector b.<br />

1-17

logo

products

Resources

Company

Terms of service
Privacy policy
Cookie policy
Cookie settings
Imprint
Change language
Made with love in Switzerland
© 2024 Yumpu.com all rights reserved

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!