C208 Roots of a Quartic Equation

Subroutine subprograms RRTEQ4 and DRTEQ4 compute the four roots of

$x4+\; ax3+\; bx2+\; cx\; +\; d\; =\; 0\; (*)$

for real coefficients a,b,c,d.

On computers other than CDC or Cray, only the double-precision version DRTEQ4 is available. On CDC and Cray computers, only the single-precision version RRTEQ4 is available.

Structure:

SUBROUTINE subprograms
User Entry Names: RRTEQ4, DRTEQ4
Obsolete User Entry Names: RTEQ4 $\equiv$ RRTEQ4
External References: RRTEQ3, DRTEQ3 (C207)

Usage:

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

    CALL tRTEQ4(A,B,C,D,Z,DC,MT)

A,B,C,D

(type according to t) Coefficients a,b,c,d in $(*)$ .

Z

( COMPLEX for $t=R$ , COMPLEX*16 for $t=D$ ) One-dimensional array of length $\ge 4$ . On exit, Z contains the roots of $(*)$ .

DC

(type according to t) On exit, DC is set to the value of the discriminant of the cubic resolvent of $(*)$ .

MT

( INTEGER) On exit, MT specifies the type of the roots:
$=\; 1:$ Four real roots in $Z(1),...,Z(4)$ ;
$=\; 2:$ Two pairs of complex conjugate roots, one pair in Z(1), Z(2), the other in Z(3), Z(4);
$=\; 3:$ Two real roots in Z(1), Z(2), and one pair of complex conjugate roots in Z(3), Z(4).

Method:

The equation is solved by the classical procedure, i.e., by solving its cubic resolvent and by combining the square roots of these solutions appropriately.

Accuracy:

Depends on the coefficients a,b,c,d. The values of $Z(1),...,Z(4)$ and of DC may be inaccurate if |DC| is very small. MT may be uncertain in such cases.
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C209