Gershgorin Circle Theorem - Application

Application

The Gershgorin circle theorem is useful in solving matrix equations of the form Ax = b for x where b is a vector and A is a matrix with a large condition number.

In this kind of problem, the error in the final result is usually of the same order of magnitude as the error in the initial data multiplied by the condition number of A. For instance, if b is known to six decimal places and the condition number of A is 1000 then we can only be confident that x is accurate to three decimal places. For very high condition numbers, even very small errors due to rounding can be magnified to such an extent that the result is meaningless.

It would be good to reduce the condition number of A. This can be done by preconditioning: A matrix P such that PA−1 is constructed, and then the equation PAx = Pb is solved for x. Using the exact inverse of A would be nice but finding the inverse of a matrix is generally very difficult.

Now, since PAI where I is the identity matrix, the eigenvalues of PA should all be close to 1. By the Gershgorin circle theorem, every eigenvalue of PA lies within a known area and so we can form a rough estimate of how good our choice of P was.

Read more about this topic:  Gershgorin Circle Theorem

Famous quotes containing the word application:

    My business is stanching blood and feeding fainting men; my post the open field between the bullet and the hospital. I sometimes discuss the application of a compress or a wisp of hay under a broken limb, but not the bearing and merits of a political movement. I make gruel—not speeches; I write letters home for wounded soldiers, not political addresses.
    Clara Barton (1821–1912)

    Courage is resistance to fear, mastery of fear—not absence of fear. Except a creature be part coward it is not a compliment to say it is brave; it is merely a loose application of the word. Consider the flea!—incomparably the bravest of all the creatures of God, if ignorance of fear were courage.
    Mark Twain [Samuel Langhorne Clemens] (1835–1910)

    The best political economy is the care and culture of men; for, in these crises, all are ruined except such as are proper individuals, capable of thought, and of new choice and the application of their talent to new labor.
    Ralph Waldo Emerson (1803–1882)