www.delorie.com/gnu/docs/calc/calc_102.html   search  
 
Buy the book!


GNU Emacs Calc 2.02 Manual

[ < ] [ > ]   [ << ] [ Up ] [ >> ]         [Top] [Contents] [Index] [ ? ]

3.7.68 Programming Tutorial Exercise 10

Taking the derivative of a term of the form x^n will produce a term like n x^(n-1). Taking the derivative of a constant produces zero. From this it is easy to see that the nth derivative of a polynomial, evaluated at x = 0, will equal the coefficient on the x^n term times n!.

(Because this definition is long, it will be repeated in concise form below. You can use M-# m to load it from there. While you are entering a Z ` Z ' body in a macro, Calc simply collects keystrokes without executing them. In the following diagrams we'll pretend Calc actually executed the keystrokes as you typed them, just for purposes of illustration.)

 
2:  5 x^4 + (x + 1)^2          3:  5 x^4 + (x + 1)^2
1:  6                          2:  0
    .                          1:  6
                                   .

  ' 5 x^4 + (x+1)^2 RET 6        C-x ( Z `  [ ] t 1  0 TAB

Variable 1 will accumulate the vector of coefficients.

 
2:  0              3:  0                  2:  5 x^4 + ...
1:  5 x^4 + ...    2:  5 x^4 + ...        1:  1
    .              1:  1                      .
                       .

   Z ( TAB         RET 0 s l x RET            M-TAB ! /  s | 1

Note that s | 1 appends the top-of-stack value to the vector in a variable; it is completely analogous to s + 1. We could have written instead, r 1 TAB | t 1.

 
1:  20 x^3 + 2 x + 2      1:  0         1:  [1, 2, 1, 0, 5, 0, 0]
    .                         .             .

    a d x RET                 1 Z )         DEL r 1  Z ' C-x )

To convert back, a simple method is just to map the coefficients against a table of powers of x.

 
2:  [1, 2, 1, 0, 5, 0, 0]    2:  [1, 2, 1, 0, 5, 0, 0]
1:  6                        1:  [0, 1, 2, 3, 4, 5, 6]
    .                            .

    6 RET                        1 + 0 RET 1 C-u v x

 
2:  [1, 2, 1, 0, 5, 0, 0]    2:  1 + 2 x + x^2 + 5 x^4
1:  [1, x, x^2, x^3, ... ]       .
    .

    ' x RET TAB V M ^            *

Once again, here are the whole polynomial to/from vector programs:

 
C-x ( Z `  [ ] t 1  0 TAB
           Z (  TAB RET 0 s l x RET M-TAB ! /  s | 1
                a d x RET
         1 Z ) r 1
      Z '
C-x )

C-x (  1 + 0 RET 1 C-u v x ' x RET TAB V M ^ *  C-x )


[ < ] [ > ]   [ << ] [ Up ] [ >> ]         [Top] [Contents] [Index] [ ? ]

  webmaster   donations   bookstore     delorie software   privacy  
  Copyright 2003   by The Free Software Foundation     Updated Jun 2003