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Unformatted text preview: Equations and Their Solutions Numerical (Scalar) Equations These are simply equations of the form f ( x ) = 0, such as one might encounter in analytic geometry or elementary calculus. A solution is a number, ˆ x , which reduces the equation to an identity, i.e., a number such that f (ˆ x ), in fact, is zero. Examples Consider the equations f ( x ) ≡ x 3 8 = 0; g ( x ) ≡ sin x cos x = 0 . In the first case we can factor and write f ( x ) = ( x 2) x 2 + 2 x + 4 = 0 and immediately conclude that x = 2 is a solution; it is the only real solution since the solutions of x 2 + 2 x + 4 = 0 are both complex. For the second equation we note that sin x cos x = √ 2 sin x 1 √ 2 cos x 1 √ 2 ! = √ 2 sin x π 4 ! since sine and cosine are both equal to 1 / √ 2 at x = π/ 4. The solutions of sin y = 0 have the form y = k π , k an arbitrary integer, so the equation g ( x ) = 0 has a solution set consisting of all numbers ( π 4 + kπ k = 0 , ± 1 , ± 2 , ... ) . In such a situation we would have to impose other conditions on x if we want the solution to be unique. In some cases, like those above, we can obtain exact forms for the solutions analytically. Often, however, this is not possible and we have 1 to solve the equation in an approximate, numerical sense. An approx imate solution method is a procedure, P, with the following property. Given an initial point, x , sufficiently near the desired solution ˆ x , P will produce a sequence { x k  k = 0 , 1 , 2 , 3 , ... } with the property that lim k →∞ x k = ˆ x. We say that the sequence { x k } converges to the solution ˆ x . The best known approximate solution method is Newton’s Method . Suppose we have a starting point x known to be somewhere close to the desired solution ˆ x and suppose that the function f ( x ) is contin uously differentiable, i.e., f ( x ) = df dx is defined and is a continuous function of x . Further, we suppose that that f ( x ) is nonzero in some neighborhood of the solution ˆ x which includes the starting point x . The definition of f ( x ) is f ( x ) = lim x → x f ( x ) f ( x ) x x . We can restate this as f ( x ) f ( x ) x x = f ( x ) + h ( x ) where h ( x ) tends to zero as x tends to x . From this we have f ( x ) = f ( x ) + f ( x ) ( x x ) + h ( x ) ( x x ) . Setting the left hand equal to zero, our original task, is the same thing as setting the right hand equal to zero. The approximation consists in not quite setting the right hand equal to zero but, rather, setting f ( x ) + f ( x ) ( x x ) = 0 ....
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 Spring '10
 ROBINSON
 Calculus, Geometry, Numerical Analysis, Equations, Scalar

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