POLYNOMIAL

POLYNOMIAL fits a polynomial curve of up to degree 9 (up to 10 coefficients) to as many as 20 data points using the method of least squares. The program builds the normal equations by accumulating power sums of the x-values into arrays, then solves the resulting symmetric matrix system using Cramer’s rule combined with a Gaussian elimination subroutine to compute determinants. It reports each coefficient C(N), the reduced chi-squared statistic CHIR (sum of squared residuals divided by the degrees of freedom P−NT), and allows the user to evaluate the fitted polynomial at arbitrary x-values. The determinant subroutine at line 1050 includes partial column pivoting to handle zero diagonal elements and flips the sign of DET when columns are swapped.

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Program Structure

The program divides into four logical phases:

1. Data entry (lines 20–250): Prompts for the number of data points P (up to 20), then collects x/y pairs into arrays U() and V().
2. Normal-equation assembly (lines 260–540): For a chosen number of coefficients NT, accumulates power sums X(N) = Σxⁿ and cross-sums Y(N) = Σyxⁿ, then fills the symmetric Gram matrix A(J,K).
3. Cramer’s rule solution (lines 550–830): Calls the determinant subroutine for the system determinant D, then iterates over each column, replacing it with Y() and re-computing the determinant to obtain each coefficient C(L) = DET/D.
4. Evaluation and iteration (lines 860–1040): Lets the user query the polynomial at arbitrary x-values, or restart with a different NT, or return to initial data entry.

Determinant Subroutine (lines 1050–1240)

The subroutine beginning at line 1050 computes the determinant of A() by upper-triangular (Gaussian) reduction, accumulating the diagonal product into DET. Partial pivoting is implemented by scanning for a nonzero element in row K of column K: if A(K,K)=0, it searches columns K through NT for a nonzero entry and swaps the entire row, negating DET for the column swap. If no nonzero pivot is found the matrix is singular and DET=0 is returned.

There is a structural anomaly: the inner elimination loops at lines 1210–1230 are followed by three NEXT clauses on line 1240 (NEXT J:NEXT I:NEXT K:RETURN). This means the outer FOR K loop begun at line 1060 is closed inside the elimination block rather than after it, so the RETURN is only ever reached from within that path. The separate GO TO 1260 at line 1200 targets a non-existent line — in practice, when K=NT the condition is true and execution falls through to line 1240’s NEXT K:RETURN, which correctly terminates. This is a deliberate (if fragile) short-circuit.

Array Dimensioning

Array	Size	Purpose
X(19)	19	Power sums Σxⁿ for n=1…2NT−1
Y(10)	10	Cross-sums Σyxⁿ
A(10,10)	10×10	Gram/normal-equation matrix
C(10)	10	Fitted polynomial coefficients
U(20)	20	Input x-values
V(20)	20	Input y-values

Note that X() is dimensioned to 19, which is exactly 2×10−1, matching the maximum NT=10 case. Similarly U() and V() hold up to 20 points as advertised.

Reduced Chi-Squared Calculation

The program computes the residual sum of squares analytically rather than by re-evaluating the polynomial at each data point. Starting from CHI = Σy² (line 470), it applies the identity:

CHI = Σy² − 2·Σ C(J)·Y(J) + Σ C(J)·C(K)·X(J+K−1)

using the already-accumulated power sums, then divides by P−NT to give the reduced chi-squared CHIR. This avoids a separate evaluation loop and reuses the stored sums efficiently.

User Interaction Flow

The navigation uses a flag variable T (initialized 0 at line 860) to distinguish between two re-entry paths. After evaluating y-values, entering 999 as x sets T=1 and jumps to the “try another N?” prompt. If the user answers Y, execution goes to line 260 (re-fit only, no re-entry of data). If the user answers N and T=1, line 910 sends execution back to line 20 for a completely fresh run. This double-use of line 870 as a shared branch target for two different exit conditions is a compact but somewhat opaque control structure.

Key BASIC Idioms

• PAUSE 0 followed by a keyboard prompt (line 250) is used to halt output until the user verifies entered data.
• PAUSE 400 (line 80) provides approximately a 5-second delay as noted in the on-screen message.
• Polynomial evaluation at lines 960–1000 uses the Horner-like accumulation of XT = XX^(L−1) rather than full Horner’s method, which is slightly less efficient but straightforward.
• The SAVE "POLYNOMIAL" LINE 10 at line 1280 stores the program with an auto-run entry point.

Limitations and Potential Issues

• Cramer’s rule requires recomputing the full determinant NT times, so for NT=10 this involves 11 calls to an O(NT³) subroutine — slow but functional for small NT.
• The matrix A() is overwritten during each determinant call (Gaussian elimination is destructive), so it must be rebuilt from X() and Y() at lines 640–700 before each Cramer substitution — the code does this correctly.
• No input validation guards against P > 20 or NT > 10, which would cause array out-of-bounds errors.
• Line 15 contains a note (“install your printer driver LOAD LINE here”) suggesting the listing was designed to optionally include a printer driver loaded between lines 15 and 20.