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Page 5-19
An alternate definition of the Hermite polynomials is
where d
n
/dx
n
= n-th derivative with respect to x. This is the definition used in
the calculator.
Examples: The Hermite polynomials of orders 3 and 5 are given by:
HERMITE(3) = ‘8*X^3-12*X’,
And HERMITE(5) = ‘32*x^5-160*X^3+120*X’.
The HORNER function
The function HORNER produces the Horner division, or synthetic division, of a
polynomial P(X) by the factor (X-a). The input to the function are the polynomial
P(X) and the number a. The function returns the quotient polynomial Q(X) that
results from dividing P(X) by (X-a), the value of a, and the value of P(a), in that
order. In other words, P(X) = Q(X)(X-a)+P(a). For example,
HORNER(‘X^3+2*X^2-3*X+1’,2) = {‘X^2+4*X+5’, 2, 11}. We could,
therefore, write X
3
+2X
2
-3X+1 = (X
2
+4X+5)(X-2)+11. A second example:
HORNER(‘X^6-1’,-5)=
{’X^5-5*X^4+25*X^3-125*X^2+625*X-3125’,-5, 15624}
i.e., X
6
-1 = (X
5
-5*X
4
+25X
3
-125X
2
+625X-3125)(X+5)+15624.
The variable VX
A variable called VX exists in the calculator’s {HOME CASDIR} directory that
takes, by default, the value of ‘X’. This is the name of the preferred
independent variable for algebraic and calculus applications. Avoid using the
variable VX in your programs or equations, so as to not get it confused with the
CAS’ VX. If you need to refer to the x-component of velocity, for example, you
can use vx or Vx. For additional information on the CAS variable see
Appendix C.
The LAGRANGE function
The function LAGRANGE requires as input a matrix having two rows and n
columns. The matrix stores data points of the form [[x
1
,x
2
, …, x
n
] [y
1
, y
2
, …,
y
n
]]. Application of the function LAGRANGE produces the polynomial
expanded from
,...2,1),()1()(*,1*
22
0
=−==
−
ne
dx
d
exHH
x
n
n
xn
n