User:Mark W. Miller

From Wikipedia, the free encyclopedia

1 Articles written or extensively rewritten
2 American Civil War
3 Math and Statistics
4 Biometrics
5 Sports
6 Newton-Raphson Method and Hessian Matrix

[edit] Articles written or extensively rewritten

[edit] American Civil War

Battle of Salineville (written before registering, editted by User:Scott Mingus)

[edit] Math and Statistics

Matrix population models (written)

[edit] Biometrics

Mark and recapture (extensively rewritten)

[edit] Sports

Aaron Smith (football player) (extensively rewritten)

[edit] Newton-Raphson Method and Hessian Matrix

NOTE TO SELF: THERE WAS A TYPO IN THE HESSIAN MATRIX IN THE BOOK. THE INVERSE OF THE HESSIAN IS CORRECT, BUT THE HESSIAN ITSELF WAS NOT.

THE HESSIAN NOW IS CORRECT BELOW.

I posted a question recently about the Hessian Matrix on the Math Talk page and received some good replies. I posted a follow-up question later in the Math and Science Question section, but then found the error in the book myself. Below deals with using the Hessian Matrix and Newton-Raphson Method to locate the minima or maxima of a function. The example in REA's Problem Solvers book "Operations Research", p. 739-740, contains an error in the Hessian Matrix.

The function in the example contains two unknowns, x₁ and x₂, and is:

4x₁² + 2x₁x₂ + 2x₂² + x₁ + x₂.

The problem is to find the values of x₁ and x₂ that provide the minima.

The matrix of partial derivatives is: f(x) = $\begin{bmatrix}8x1 + 2x2 + 1 \\2x1 + 4x2 + 1\end{bmatrix}$ = 0.

The Hessian Matrix is:

H(x) = $\begin{bmatrix}8&2\\2&4\end{bmatrix}$ ;

Not:

H(x) = $\begin{bmatrix}8&2\\8&4\end{bmatrix}$ as in the book.

The third matrix is indeed the inverse of the Hessian Matrix, and is given correctly in the book as:

H^-1(x) = (1/14) * $\begin{bmatrix} 2&-1\\-1&4\end{bmatrix}$ .

The interative formula used to find the values of x₁ and x₂ corresponding to the minima of the original function is:

x⁽ⁿ⁺¹⁾ = x⁽ⁿ⁾ - H^-1(x) * f(x);

and gives the values x⁽ⁿ⁺¹⁾₁ = -1/14 and x⁽ⁿ⁺¹⁾₂ = -3/14.

I understand, at least mechanically, the substitutions and algebra used to arrive at those answers given the formula for x⁽ⁿ⁺¹⁾.

Note that:

H(x) * H^-1(x) = $\begin{bmatrix} 1&0\\0&1\end{bmatrix}$ .

Note that the eigenvalues of the Hessian Matrix are 8.8284271 and 3.1715729, which I think corresponds to a minima, which is what the REA "Operations Research" book said was the case with this particular function.

-- Mark W. Miller 06:49, 27 November 2005 (UTC)

Retrieved from "http://en.wikipedia.org../../../m/a/r/User%7EMark_W._Miller_cde6.html"

User:Mark W. Miller

From Wikipedia, the free encyclopedia

Contents

[edit] Articles written or extensively rewritten

[edit] American Civil War

[edit] Math and Statistics

[edit] Biometrics

[edit] Sports

[edit] Newton-Raphson Method and Hessian Matrix

Views

Navigation

Search