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```SUBROUTINE ARAT

PURPOSE
CALCULATE BEST RATIONAL APPROXIMATION OF A DISCRETE
FUNCTION IN THE LEAST SQUARES SENSE

USAGE
CALL ARAT(DATI,N,WORK,P,IP,IQ,IER)

DESCRIPTION OF PARAMETERS
DATI  - TWODIMENSIONAL ARRAY WITH 3 COLUMNS AND N ROWS
THE FIRST COLUMN MUST CONTAIN THE GIVEN ARGUMENTS,
THE SECOND COLUMN THE GIVEN FUNCTION VALUES AND
THE THIRD COLUMN THE GIVEN WEIGHTS IF ANY.
IF NO WEIGHTS ARE TO BE USED THEN THE THIRD
COLUMN MAY BE DROPPED , EXCEPT THE FIRST ELEMENT
WHICH MUST CONTAIN A NONPOSITIVE VALUE
N	 - NUMBER OF NODES OF THE GIVEN DISCRETE FUNCTION
WORK  - WORKING STORAGE WHICH IS OF DIMENSION
(IP+IQ)*(IP+IQ+1)+4*N+1 AT LEAST.
ON RETURN THE VALUES OF THE NUMERATOR ARE CONTAINED
IN WORK(N+1) UP TO WORK(2*N), WHILE THE VALUES OF
THE DENOMINATOR ARE STORED IN WORK(2*N+1) UP TO
WORK(3*N)
P	 - RESULTANT COEFFICIENT VECTOR OF DENOMINATOR AND
NUMERATOR. THE DENOMINATOR IS STORED IN FIRST IQ
LOCATIONS, THE NUMERATOR IN THE FOLLOWING IP
LOCATIONS.
COEFFICIENTS ARE ORDERED FROM LOW TO HIGH.
IP	 - DIMENSION OF THE NUMERATOR	(INPUT VALUE)
IQ	 - DIMENSION OF THE DENOMINATOR (INPUT VALUE)
IER	 - RESULTANT ERROR PARAMETER
IER =-1 MEANS FORMAL ERRORS
IER = 0 MEANS NO ERRORS
IER = 1,2 MEANS POOR CONVERGENCE OF ITERATION
IER IS ALSO USED AS INPUT VALUE
A NONZERO INPUT VALUE INDICATES AVAILABILITY OF AN
INITIAL APPROXIMATION STORED IN P

REMARKS
THE COEFFICIENT VECTORS OF THE DENOMINATOR AND NUMERATOR
OF THE RATIONAL APPROXIMATION ARE BOTH STORED IN P
STARTING WITH LOW POWERS (DENOMINATOR FIRST).
IP+IQ MUST NOT EXCEED N, ALL THREE VALUES MUST BE POSITIVE.
SINCE CHEBYSHEV POLYNOMIALS ARE USED AS FUNDAMENTAL
FUNCTIONS, THE ARGUMENTS SHOULD BE REDUCED TO THE INTERVAL
(-1,1). THIS CAN ALWAYS BE ACCOMPLISHED BY MEANS OF A LINEAR
TRANSFORMATION OF THE ORIGINALLY GIVEN ARGUMENTS.
IF A FIT IN OTHER FUNCTIONS IS REQUIRED, CNP AND CNPS MUST
BE REPLACED BY SUBROUTINES WHICH ARE OF ANALOGOUS DESIGN.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
APLL, APFS, FRAT, CNPS, CNP
CNP IS REQUIRED WITHIN FRAT

METHOD
THE ITERATIVE SCHEME USED FOR CALCULATION OF THE
APPROXIMATION IS REPEATED SOLUTION OF THE NORMAL EQUATIONS
WHICH ARE OBTAINED BY LINEARIZATION.
A REFINED TECHNIQUE OF THIS LINEAR LEAST SQUARES APPROACH
IS USED WHICH GUARANTEES THAT THE DENOMINATOR IS FREE OF
ZEROES WITHIN THE APPROXIMATION INTERVAL.
FOR REFERENCE SEE
D.BRAESS, UEBER DAEMPFUNG BEI MINIMALISIERUNGSVERFAHREN,
COMPUTING(1966), VOL.1, ED.3, PP.264-272.
D.W.MARQUARDT, AN ALGORITHM FOR LEAST-SQUARES ESTIMATION
OF NONLINEAR PARAMETERS,
JSIAM(1963), VOL.11, ED.2, PP.431-441.
```