 Nonlinear EquationsHeinz Rutishauser Chapter Chapter 4 in Lectures on Numerical Mathematics, 1990, pp 77-102 from Springer Abstract: 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 (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-3468-5_4 Ordering information: This item can be ordered from DOI: 10.1007/978-1-4612-3468-5_4 Access Statistics for this chapter More chapters in Springer Books from Springer |
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