C207 Roots of a Cubic Equation

Subroutine subprograms RRTEQ3 and DRTEQ3 compute the three roots of

$x3+\; rx2+\; sx\; +\; t\; =\; 0\; (*)$

for real coefficients r, s, t.

On computers other than CDC or Cray, only the double-precision version DRTEQ3 is available. On CDC and Cray computers, only the single-precision version RRTEQ3 is available.

Structure:

SUBROUTINE subprograms
User Entry Names: RRTEQ3, DRTEQ3
Obsolete User Entry Names: RTEQ3 $\equiv$ RRTEQ3

Usage:

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

    CALL tRTEQ3(R,S,T,X,D)

R,S,T

(type according to t) Coefficients r,s,t in $(*)$ .

X

(type according to t) One-dimensional array of length $\ge 3$ . On exit, X is set as described below.

D

(type according to t) On exit, D is set to the value of the discriminant of $(*)$ :
$>\; 0:$ One real root X(1) and two complex conjugate roots $X(2)+iX(3)$ , $X(2)-iX(3)$ ;
$=\; 0:$ Three real roots X(1), X(2), X(3), where at least $X(2)=X(3)$ ;
Three distinct real roots X(1), X(2), X(3).

Method:

The classical method of Tartaglia-Vieta is used. In certain cases, the solutions are improved by Newton iteration.

Accuracy:

Depends on the coefficients r,s,t. The values of X(1), X(2), X(3) and of D may be inaccurate if |D| is very small, even in the case of ``exact'' coefficients.
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C208