Gauss Calculations

This program implements Gaussian elimination with partial pivoting to solve systems of linear equations, translated from a Fortran IV mathematical routine. It accepts an N×N coefficient matrix A and a vector B (up to 20×20), prompts for an epsilon threshold for near-singular detection, and outputs the solution vector X and the determinant of A. Partial pivoting is performed by searching each column for the largest absolute pivot element and swapping rows as needed, with the determinant sign adjusted for each interchange. A subroutine at line 1000 displays the input matrix and vector before elimination begins, and lines 1500–1600 contain an unreferenced debug-style routine for printing intermediate row values. A self-contained test case using hardcoded DATA is defined at line 2000 but is never reached during normal execution flow.

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Program Structure

The program is organized into several distinct phases:

1. Initialization and input (lines 10–90): Splash screen, epsilon input with default fallback, and entry of matrix A and vector B.
2. Matrix display subroutine (lines 1000–1050): Prints the full A matrix and B vector before elimination.
3. Gaussian elimination with partial pivoting (lines 200–520): The main forward-elimination loop.
4. Back substitution (lines 570–670): Solves for each X(I) from X(N) down to X(1).
5. Output (lines 680–730): Prints the solution vector and determinant.
6. Error handler (line 1900): Triggered when the pivot is below epsilon, indicating no unique solution.
7. Dead code (lines 1500–1600 and 2000–2070): An unreferenced debug print routine and a hardcoded 2×2 test case that are never reached in normal execution.

Algorithm: Gaussian Elimination with Partial Pivoting

The elimination loop (lines 210–520) searches each column K for the row with the largest absolute value pivot (lines 260–300), then swaps that row with row K if necessary (lines 330–420). This is classic partial pivoting, which improves numerical stability. The determinant sign is negated at line 340 for each row swap. The inner elimination (lines 450–510) computes a multiplier FACTOR = A(I,K)/PIVOT and subtracts the appropriate multiple of the pivot row from each lower row, also updating the B vector. Zeros in the eliminated positions are not explicitly stored, consistent with the original Fortran comment at line 440.

Back substitution (lines 580–670) starts with X(N)=B(N)/A(N,N) and works upward, accumulating dot products with already-solved X values.

The determinant is accumulated in lines 120 and 690 by multiplying the diagonal elements of the upper-triangular result, with sign corrections from row swaps.

Notable Techniques and Idioms

• POKE 23692,255 at lines 60 and 100 suppresses the scroll prompt (“scroll?”) by resetting the line counter, a standard idiom for programs with heavy screen output.
• Epsilon is accepted as a string (E$) at line 20, allowing an empty ENTER to trigger the default of 1E-05 at line 30 via a string-empty check, rather than requiring a special sentinel numeric value.
• VAL E$ at line 40 converts the user’s string input to a numeric epsilon, a compact idiom for optional numeric input.
• The variable NLESS1=N-1 (line 130) pre-computes a loop bound used in multiple places, a minor efficiency measure inherited from Fortran practice.
• PAUSE 0 at line 730 followed by GO TO 10 implements a keypress-to-restart idiom.
• The matrix display subroutine reuses the variable names A and B as loop counters (lines 1010–1030), which shadows the arrays A() and B() within that loop. In Sinclair BASIC, simple variables and array variables share the same name space but are stored separately, so A (scalar) and A() (array) coexist without conflict — this is intentional and works correctly.

Bugs and Anomalies

Line	Issue
310	IF ABS ((PIVOT) <=EPSLN) — the closing parenthesis is misplaced; this parses as ABS(PIVOT <= EPSLN), evaluating the boolean comparison first and then taking its absolute value (always 0 or 1), rather than checking ABS(PIVOT) <= EPSLN. The intended singular-matrix guard will not function correctly for most values of EPSLN.
1500–1600	This subroutine is defined and has a RETURN but is never called from anywhere in the program. It appears to be a debug-printing routine left over from development.
2000–2070	A hardcoded 2×2 self-test using DATA and RESTORE that jumps to line 100. It is never reached in normal execution flow and appears to be an abandoned test harness.
1900	GO TO 1 targets a non-existent line; execution will fall through to the next available line (line 5), effectively restarting the program. This is a known BASIC technique rather than a defect.

Variable Summary

Variable	Role
A(20,20)	Coefficient matrix
B(20)	Right-hand side vector; modified in-place during elimination
X(20)	Solution vector
N	Number of equations/unknowns
EPSLN	Singularity threshold (default 1E-05)
PIVOT	Current pivot element
FACTOR	Row multiplier during elimination
DET	Running determinant accumulator
L	Row index of best pivot candidate
TEMP	Swap temporary for row interchange
NLESS1, KPLUS1, IPLUS1	Pre-computed index offsets