SUBROUTINE PRBM

PURPOSE
   TO CALCULATE ALL REAL AND COMPLEX ROOTS OF A GIVEN
   POLYNOMIAL WITH REAL COEFFICIENTS.

USAGE
   CALL PRBM (C,IC,RR,RC,POL,IR,IER)

DESCRIPTION OF PARAMETERS
   C	  - INPUT VECTOR CONTAINING THE COEFFICIENTS OF THE
	    GIVEN POLYNOMIAL. COEFFICIENTS ARE ORDERED FROM
	    LOW TO HIGH. ON RETURN COEFFICIENTS ARE DIVIDED
	    BY THE LAST NONZERO TERM.
   IC	  - DIMENSION OF VECTORS C, RR, RC, AND POL.
   RR	  - RESULTANT VECTOR OF REAL PARTS OF THE ROOTS.
   RC	  - RESULTANT VECTOR OF COMPLEX PARTS OF THE ROOTS.
   POL	  - RESULTANT VECTOR OF COEFFICIENTS OF THE POLYNOMIAL
	    WITH CALCULATED ROOTS. COEFFICIENTS ARE ORDERED
	    FROM LOW TO HIGH (SEE REMARK 4).
   IR	  - OUTPUT VALUE SPECIFYING THE NUMBER OF CALCULATED
	    ROOTS. NORMALLY IR IS EQUAL TO IC-1.
   IER	  - RESULTANT ERROR PARAMETER CODED AS FOLLOWS
	     IER=0  - NO ERROR,
	     IER=1  - SUBROUTINE PQFB RECORDS POOR CONVERGENCE
		      AT SOME QUADRATIC FACTORIZATION WITHIN
		      50 ITERATION STEPS,
	     IER=2  - POLYNOMIAL IS DEGENERATE, I.E. ZERO OR
		      CONSTANT,
		      OR OVERFLOW IN NORMALIZATION OF GIVEN
		      POLYNOMIAL,
	     IER=3  - THE SUBROUTINE IS BYPASSED DUE TO
		      SUCCESSIVE ZERO DIVISORS OR OVERFLOWS
		      IN QUADRATIC FACTORIZATION OR DUE TO
		      COMPLETELY UNSATISFACTORY ACCURACY,
	     IER=-1 - CALCULATED COEFFICIENT VECTOR HAS LESS
		      THAN THREE CORRECT SIGNIFICANT DIGITS.
		      THIS REVEALS POOR ACCURACY OF CALCULATED
		      ROOTS.

REMARKS
   (1) REAL PARTS OF THE ROOTS ARE STORED IN RR(1) UP TO RR(IR)
       AND CORRESPONDING COMPLEX PARTS IN RC(1) UP TO RC(IR).
   (2) ERROR MESSAGE IER=1 INDICATES POOR CONVERGENCE WITHIN
       50 ITERATION STEPS AT SOME QUADRQTIC FACTORIZATION
       PERFORMED BY SUBROUTINE PQFB.
   (3) NO ACTION BESIDES ERROR MESSAGE IER=2 IN CASE OF A ZERO
       OR CONSTANT POLYNOMIAL. THE SAME ERROR MESSAGE IS GIVEN
       IN CASE OF AN OVERFLOW IN NORMALIZATION OF GIVEN
       POLYNOMIAL.
   (4) ERROR MESSAGE IER=3 INDICATES SUCCESSIVE ZERO DIVISORS
       OR OVERFLOWS OR COMPLETELY UNSATISFACTORY ACCURACY AT
       ANY QUADRATIC FACTORIZATION PERFORMED BY
       SUBROUTINE PQFB. IN THIS CASE CALCULATION IS BYPASSED.
       IR RECORDS THE NUMBER OF CALCULATED ROOTS.
       POL(1),...,POL(J-IR) ARE THE COEFFICIENTS OF THE
       REMAINING POLYNOMIAL, WHERE J IS THE ACTUAL NUMBER OF
       COEFFICIENTS IN VECTOR C (NORMALLY J=IC).
   (5) IF CALCULATED COEFFICIENT VECTOR HAS LESS THAN THREE
       CORRECT SIGNIFICANT DIGITS THOUGH ALL QUADRATIC
       FACTORIZATIONS SHOWED SATISFACTORY ACCURACY, THE ERROR
       MESSAGE IER=-1 IS GIVEN.
   (6) THE FINAL COMPARISON BETWEEN GIVEN AND CALCULATED
       COEFFICIENT VECTOR IS PERFORMED ONLY IF ALL ROOTS HAVE
       BEEN CALCULATED. IN THIS CASE THE NUMBER OF ROOTS IR IS
       EQUAL TO THE ACTUAL DEGREE OF THE POLYNOMIAL (NORMALLY
       IR=IC-1). THE MAXIMAL RELATIVE ERROR OF THE COEFFICIENT
       VECTOR IS RECORDED IN RR(IR+1).

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   SUBROUTINE PQFB     QUADRATIC FACTORIZATION OF A POLYNOMIAL
		       BY BAIRSTOW ITERATION.

METHOD
   THE ROOTS OF THE POLYNOMIAL ARE CALCULATED BY MEANS OF
   SUCCESSIVE QUADRATIC FACTORIZATION PERFORMED BY BAIRSTOW
   ITERATION. X**2 IS USED AS INITIAL GUESS FOR THE FIRST
   QUADRATIC FACTOR, AND FURTHER EACH CALCULATED QUADRATIC
   FACTOR IS USED AS INITIAL GUESS FOR THE NEXT ONE. AFTER
   COMPUTATION OF ALL ROOTS THE COEFFICIENT VECTOR IS
   CALCULATED AND COMPARED WITH THE GIVEN ONE.
   FOR REFERENCE, SEE J. H. WILKINSON, THE EVALUATION OF THE
   ZEROS OF ILL-CONDITIONED POLYNOMIALS (PART ONE AND TWO),
   NUMERISCHE MATHEMATIK, VOL.1 (1959), PP.150-180.