246 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Suppose we wish to solve for E=.14, which represents an expected
return of 14%. Plugging .14 into the matrix for E and putting in zeros for the
variables L 1 and L 2 in the first two rows to complete the matrix gives us a
matrix of:
X 1 X 2 X 3 X 4 L 1 L 2 | Answer.095 .13 .21 .085 0 0 | .14
1 1 1100| 1
.1 −.0237 .01 0 .095 1 | 0
−.0237 .25 .079 0 .13 1 | 0
.01 .079 .4 0 .21 1 | 0
0 0 0 0 .085 1 | 0By solving the matrix we will solve the N+2 unknowns in the N+ 2
equations.
Solutions of Linear Systems Using Row-Equivalent Matrices
ROW-EQUIVALENT MATRICES
Apolynomialis an algebraic expression that is the sum of one or more
terms. A polynomial with only one term is called amonomial; with two
terms abinomial; with three terms atrinomial.Polynomials with more
than three terms are simply called polynomials. The expression 4∗A^3 +A^2 +
A+2 is a polynomial having four terms. The terms are separated by a plus
(+) sign.
Polynomials come in differentdegrees.The degree of a polynomial is
the value of the highest degree of any of the terms. The degree of a term
is the sum of the exponents on the variables contained in the term. Our
example is a third-degree polynomial since the term 4∗A^3 is raised to the
power of 3, and that is a higher power than any of the other terms in the
polynomial are raised to. If this term read 4∗A^3 ∗B^2 ∗C, we would have a
sixth-degree polynomial since the sum of the exponents of the variables (3
- 2 +1) equals 6.
A first-degree polynomial is also called alinear equation, and it graphs
as a straight line. A second-degree polynomial is called aquadratic, and it
graphs as a parabola. Third-, fourth-, and fifth-degree polynomials are also
calledcubics, quartics, andquintics, respectively. Beyond that there aren’t
any special names for higher-degree polynomials. The graphs of polynomials
greater than second degree are rather unpredictable. Polynomials can have
any number of terms and can be of any degree. Fortunately, we will be
working only with linear equations, first-degree polynomials here.