FIGURING THINGS OUT

It is pleasant to see a company which is not afraid to cover a specialised area in the Sinclair market. University Software produces five cassette tapes for students at A level or university standard. The programs are available for the 16K ZX-81 and the 16K Spectrum. Tape one introduces matrix operations. The second explains polynomials. It includes quadratic equations, Newton-Raphson and half-interval search methods.

Tape three deals with integration, using Simpson's and trapezoidal rules.

The fourth tape covers regression. The program can deal with up to 20 independent variables, with standard errors, and also illustrates interpolation.

Tape five concerns linear programming and is capable of handling up to 20 variables and the same number of constraints.

All programs can be bought separately, or as a package costing £30. The first three tapes cost £5.95 and the fourth and fifth £6.95. They arc available from University Software, 45C Sloane Street, London SW1X 9LU.

COLLEGE COMUTING

Nick Pearce goes back to college and takes a look at the library of advanced mathematics software from University Software.

University Software have produced five cassettes which comprise their library of advanced mathematics. They are designed to handle complex problems in various branches of mathematics.

These are utility, rather than teaching, programs, although instructions printed on the cassettes covers are 'intended to introduce the nonspecialist to the theory'. In general, the instructions are adequate for this purpose.

Matrix Operation. This program is designed to handle the usual matrix operations of Inversion, Multiplications, Addition, Subtraction and Scalar Multiplications. The operation required is selected from a menu at the start of the program. The user dimensions the matrices and enters the values of the matrices row by row. The program will not accept invalid instructions at this stage. For example, if you select the inversion operation the program will not accept a matrix with an unequal number of rows and columns (which cannot be inverted) and prompts the user for new matrix dimensions. Whilst there seems no limit to the matrix dimensions this program will accept, a 20 by 20 matrix is inverted in a little under seven minutes.

The program works well and data input is logically organised. However, I would have liked the option to alter individual values within a matrix; as the program stands, a mis-type during data entry necessitates re typing of the complete matrix. On the reverse side of the cassette is Determinants, a program which computes the determinant by means of converting a matrix into a triangular matrix by the appropriate row transformations.

Polynomials. This program also severely tested my knowledge of such things. A polynomial is an expression of the form:

F(x) = Cn Xn + Cn ,Xn 1 + ... + C2X2 + C,X + C0

where C n ... C0 are coefficients and n is the degree of the polynomial. The program calculates the roots of a polynomial expression, ie the values of x which satisfy F(x) = 0 (real roots only).

The program employs three different methods. If the expression has two degrees the familiar formula for solving quadratic equations is used; for higher degree polynomials interactive methods are used, either the Newton-Raphson method (which I recall) or the Half-Interval Search method (which I don't).

On the B side is Plot of Polynomials, a program which plots the polynomial equation between given limits. This works very well, you input the degree and coefficients of the polynomial as before, and the range over which it is to be plotted. The program scales the axes and plots the equation accordingly, and also labels the extreme values of the axes. If a root is found the value of the root is printed at the point of intersection.

Integration. On more familiar territory now, I can clearly recollect counting up squares to calculate the area under curves. This program evaluates the integral of functions between given limits by Simpson's and Trapezoidal rules. Two functions can be integrated simultaneously, allowing the area between two curves to be evaluated.

Again, a program to plot the function is given on the B side. This plots two functions between given intervals and cross-hatches the area between them. To evaluate the integrals the A side has to be used.

Regression. On even more familiar ground, regression analysis is a technique I have had occasion to use recently. I devised for myself a short program for the ZX81; it worked adequately but could only handle one independent and one dependent variable. University Software's program is somewhat more elaborate and will solve a 'multivariate' linear regression problem.

The user enters the number of observations and the number of independent variables. For each observation the values of the associated variables are entered. Data entry is a little tedious, and again there is no way to correct a mis-typed figure if you make a mistake you have to re-type from scratch. The program can deal, with both exponential and logarithmic regressions.

The program calculates the equation coefficients, and gives the values of r2, corrected r2 (r2 adjusted for the degrees of freedom), the F-statistic, standard error of regression, Durbin Watson statistic, and the t-statistic. Side B plots the regression line together with the numbered data points for a 'bivorate' regression - the sort of regression I understand. It gives the slope and intercept of the fitted line and the standard error and r2 of the regression.

Linear Programming. This is where things started to get difficult again. The cassette is certainly no substitute for a good textbook on the subject, but having mastered the theory it can be of considerable assistance in the solution of linear programming problems.

This optimization program is capable of handling any sort of linear programming problem with up to 20 variables and 20 constraints. After data entry, the original form of the problem (the primal) is displayed together with the solution - or with a report that the problem is either unfeasible or unbounded. A second display gives the canonical equivalent of the primal, its solution and the variable relationships or the primal and canonical, A third displays the dual and its solution.

Side B deals with simultaneous equations.

All of these University Software programs run automatically once loaded, and prompts to guide the user are given on screen. The programs are all written in 8ASIC and can be listed, enabling the user to see how they work, or modify them. Output displays can be copied onto a printer.

These are not teaching programs, but they take the tedium, and the human error, out of mathematical problem solving and will be a valuable aid to the serious user and student alike. They all perform well have obviously been carefully prepared.

University Software also offer to prepare more specialised programs to order.

University Software is at 45/c Sloane Street, London SW1X 9LU. Matrix Operation, Polynomials and integration cost £6.95 each; Regression and Linear Programming are priced at £7.95 each.