Maths Master



         Colin Carruthers and a Spectrum teach you

         a thing or two about quadratic equations.



Are you bored solving systems of equations with five

unknowns? Want a hand to invert a couple of matrices?

Shouldn't you check that difficult integration problem

you've just spent four hours doing?

  Well, the computer's ability to plough through tedious

calculations at high speed is used in this program to

provide a useful maths package.

  The program will run on both the 16K and 48K Spectrums,

but since it is written entirely in Basic it should be

quite a simple task to implement the package on other

micros.

  As far as possible, each part of the program has been

made self-contained enabling the individual to just type

in the routines he or she requires. However, the Matrix

Operations section demands that the System of Equations

routines be present - this is due to the fact that matrix

inversion and solving systems of equations can be done by

similar techniques and therefore have common program

blocks. In any case, the program must have the menu, input

a number and hit any key routines - see figure 1.



  Figure 1 Program breakdown

   500- 580: Input a number routine. Extensive use is made

             of this routine, so it is placed near the

             beginning to speed execution.

  1000-1180: Menu.

  2000-2970 System of Equations.

  3000-3500: Quadratic Equation.

  4000-4500: Equation of third degree.

  6000-6650: Matrix Determinant and Inverse.

  7000-7500: Simpson's Rule.

  9000-9030: Hit any Key to Return.



There now follows a brief description of each part of the

program and examples of the kind of mathematical problem

they solve. These examples can be used to check that the

routines have been typed in correctly.

  System of Equations: Solved using Gaussian Elimination,

each problem can have a maximum of five equations and five

unknowns. The coefficients are held within a two-dimen-

sional array - called "a". The user is prompted for each

coefficient of x in turn, with the whole array of values

shown on the screen at all times to enable checking.

Ex (n=3)      x1 + 3x2 - 4x3 = -11

             2x1 -  x2 + 3x3 =  10

             4x1 +  x2 - 2x3 =   3

This has the solution x1 = 2, x2 = -3 and x3 = 1.

  Quadratic equation: The roots are found using the classic

formula:



                 -b±  b² - 4ac

                 x =       2a

This routine allows for both real and imaginary roots.

Ex 1 x² - 3x + 2 = 0 gives x = or 2

Ex 2 x² 6x + 10 = 0 gives x = 3+/- i

Ex 3 x² - 6x + 9 = 0 gives x = 3

(double root?

[Sic, sic, sic and sic! Allow me, as a former employee of a

publisher, to bewail the horrendous state of scientific and

mathematical knowledge among type-setters and, more impor-

tantly, correctors. They'd never have allowed themselves to

get away with such nonsense if the subject had been, say,

the arts, or gardening, or even celebrity gossip. But 

maths? That's just for nerds, and they don't care. Bah and

double bah.]



 Roots of a polynominal



  Equation of third degree: This routine gives the roots of

a polynominal with a term in x³. Again, imaginary roots are

catered for, giving four types of possible solution.

Ex1 x³ - 6x² + 11x - 6 = 0

                          gives x = 1,2,3

Ex2 x³ - 3x² -[sic! another error!] 3x - 1 = 0

                          gives x = 1,1,1

Ex3 -x³ - 9x² + 81x + 729 = 0

                        gives x = 9,-9,-9

Ex4 x³ - 5x² + 7x + 13 = 0

                   gives x = -1, 3 +/- 2i

  Matrix Operations: The determinant of the given square

matrix is calculated and displayed. Assuming that this is

non-zero, the inverse is computed using Gaussian Elimina-

tion. A matrix with zero determinant has no inverse. The

main "invert" routine is the same as that for the System of

Equations.

Ex

(n=3)



3  1  2

2  1  0

2  1  1

              has determinant 1

                    and inverse



                        1  1 -2

                       -2 -1  4

                        0 -1  1



Note that only real matrix elements are allowed.

  Simpson's Rule: The function entered must be a valid

expression in 'x', for example 'y = 3x + 2' must be entered

as:

              y = 3*x+2

  Also, functions such as Sin, Tan or Ln must be entered as

single-stroke key words. Any invalid expression typed in

response to the prompt will result in an error at line

7100, statement 3. If this should happen, simply type:

        GO TO 7000

and re-type the expression correctly.

Ex

y = 3*x + 2

lower x = 0

upper x = 4

samples = 10



 Value of the integral



As can be seen by looking at figure 2, the value of the

integral - or shaded part of the graph - should be 32.



            ^       / 

            |     |/ y=3x+2

          14+     +

            |    /|

            |   /#|

            |  /##|

            | /###|

           2|/####|

       -----+-----+----

           /|#####|

 ---------+-0-----+------->

         /  |     |4

        /   |     |

       /    |

            |

  Figure 2. Area of triangle + rectangle.

  = ½ x 4 x 10 + 2 x 4

  = 20 + 12

  = 32