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Nonlinear Equations

  • Heinz Rutishauser

Abstract

To introduce the subject, we consider a few examples of nonlinear equations:
$${x^3} + x + 1 = 0$$
is an algebraic equation; there is only one unknown, but it occurs in the third power. There are three solutions, of which two are conjugate complex.
$$2x - \tan x = 0$$
is a transcendental equation. Again, only one unknown is present, but now in a transcendental function. There are denumerably many solutions.
$$\sin x + 3 \cos x = 2$$
is a transcendental equation only in an unessential way, since it can be transformed at once into a quadratic equation for eix. While there are infinitely many solutions, they can all be derived from two solutions through addition of multiples of 2π.
$${x^3} + {y^2} + 5 = 0$$
$$2x + {y^3} + 5y = 0$$
is a system of two nonlinear algebraic equations in two unknowns x and y. It can be reduced to one algebraic equation of degree 9 in only one unknown. This latter equation has nine solutions which generate nine pairs of numbers (x i ,y i ), i = 1,…, 9, satisfying the given system. (There are fewer if only real x,y are admitted.)

Keywords

Algebraic Equation Nonlinear Equation Reconstruction Error Transcendental Equation Nonlinear Algebraic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Boston 1990

Authors and Affiliations

  • Heinz Rutishauser

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