ROOTS: quadratic equations and cubic equations (H)

• solve

  quadratic equations

     A*x² + B*x + C = 0:
  • real_roots = ROOTS(ABCvector, quadratic_roots [, *errorLabel])
  • real_roots = number of real roots found
  • REAL :: ABCvector(3), quadratic_roots(2)
  • ABCvector = (A, B, C)
  • optional: *errorLabel ⇒ on error jump to label
  • Example: x² - x + 6 = 0
    • DIMENSION ABC(3), sqare_roots(2) ! define the 2 vectors
    • ABC = ( 1, -1, 6 )
    • found = ROOTS( ABC, sqare_roots ) ! sets real_roots=2 and sqare_roots = (3, -2)
• solve

  cubic equations

     A*x³ + B*x² + C*x + D = 0:
  • real_roots = ROOTS(ABCDvector, cubic_roots [, zero_discriminant [, *errorLabel]])
  • real_roots = number of real roots found
  • DIMENSION ABCDvector(4), cubic_roots(3)
  • ABCDvector = (A, B, C, D)
  • optional: zero_discriminant (default 1E-10)
    ROOTS uses Cardano"s method. There will be 3 real roots if discriminant is less than this value
  • optional: *errorLabel ⇒ on error jump to label
  • in 1225

    Leonardo of Pisa

    found the root of
    x³ + 2*x² + 10*x - 20 = 0
    to be 1 .368808107. Nobody knows by what method he achieved that. HicEst does it by
    • ABCD = (1, 2, 10, -20)
    • found = ROOTS(ABCD, cubic_roots, 1E-14, *999)
    • ! found=1 and cubic_roots(1) = 1.368808107

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