E201 Least Squares Polynomial Fit

Subroutine subprograms RLSQPM and DLSQPM fit a polynomial

$p$_m(x) = ∑_j=0^ma_jx^j

of degree m to n equally-weighted data points ($x$_i,y_i ). The calculated coefficients $a$_j are such that

$S$_m² = ∑_i=1ⁿ(y_i-p_m(x_i))² = min.

Subroutine subprograms RLSQP1 and DLSQP1 fit a straight line $p$₁(x) = a₀+a₁x to n such points.

Subroutine subprograms RLSQP2 and DLSQP2 fit a parabola $p$₂(x) = a₀+a₁x+a₂x² to n such points.

An estimate $s\; =S$_m²/(n-m-1)

of the standard deviation $\sigma$ is calculated.

On CDC and Cray computers, the double-precision versions DLSQPM, DLSQP1 and DLSQP2 are not available.

Structure:

SUBROUTINE subprograms
User Entry Names: RLSQPM, RLSQP1, RLSQP2, DLSQPM, DLSQP1, DLSQP2
External References: RVSET, DVSET, DVSUM, DVMPY (F002), DSEQN (F012)

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    CALL tLSQPM(N,X,Y,M,A,SD,IFAIL)
    CALL tLSQP1(N,X,Y,A0,A1,SD,IFAIL)
    CALL tLSQP2(N,X,Y,A0,A1,A2,SD,IFAIL)

N

( INTEGER) Number n of data points.

X

(type according to t) One-dimensional array of length $\ge N$ . On entry, X(i) contains the abscissas $x$_i, (i=1,2,...,n) .

Y

(type according to t) One-dimensional array of length $\ge N$ . On entry, Y(i) contains the ordinates $y$_i, (i=1,2,...,n) .

M

( INTEGER) Degree m of the polynomial to be fitted.

A

(type according to t) One-dimensional array of dimension (0:d), where $d\; \ge M$ . Contains, on exit, in A(j) the coefficients $a$_j, (j = 0,1,...,m) .

A0,A1,A2

(type according to t) Contain, on exit, the coefficients $a$₀ , $a$₁ for $p$₁(x)=a₀+a₁x or $a$₀,a₁,a₂ for $p$₂(x)=a₀+a₁x+a₂x² , respectively.

SD

(type according to t) Contains, on exit, the estimate s.

IFAIL

( INTEGER) Error flag.
$=\; 0:$ Normal case,
$=\; 1:$ $N\; \le 1$ or or $M\; \ge N$ or $M>20$ ,
$=\; -1:$ The matrix of normal equations is numerically singular.

Method:

The normal equations are solved. On computers other than CDC or Cray, double-precision mode arithmetic is used internally for RLSQPM, RLSQP1 and RLSQP2.

Notes:

Meaningful results can usually be obtained only for small values of m (typically ).

References:

1. D.H. Menzel, Fundamental formulas of physics, v. 1, (Dover, New York 1960) 116--122.

E208