By J.H. Hubbard, B.H. West
It is a corrected 3rd printing of the 1st a part of the textual content Differential Equations: A Dynamical structures technique written via John Hubbard and Beverly West. The authors' major emphasis during this publication is on traditional differential equations. The publication is well suited for top point undergraduate and graduate scholars within the fields of arithmetic, engineering, and utilized arithmetic, in addition to the existence sciences, physics and economics. conventional classes on differential equations specialize in thoughts resulting in recommendations. but such a lot differential equations don't admit options that are written in user-friendly phrases. The authors have taken the view differential equations defines features; the item of the idea is to appreciate the habit of those features. The instruments the authors use contain qualitative and numerical equipment in addition to the conventional analytic tools. The significant other software program, MacMath, is designed to deliver those notions to lifestyles.
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Additional resources for Differential equations: a dynamical systems approach 1
The motivation given for minimizing /(y) instead of /(x) was the observation that if E makes K(H) < K(H), then the rate of convergence of the method of steepest descent is greater for /(y) than for /(x). Since the rate of convergence of the conjugate gradient method also depends on the spectral condition number of the Hessian, preconditioning is just as relevant for this method. We shall begin this section with some general remarks concerning preconditioning matrices and show that the class of so-called stationary iterative methods for solving the system Hx = b is a source of such matrices.
50) is very useful and motivates our use of the spectral condition number K(H) to assess the rate of convergence. 50) may be quite pessimistic, hiding the true rate of convergence. For further reading on this topic see Stewart (1975), Axelsson (1976), Jennings (1977), Greenbaum (1979), and also van der Vorst and van der Sluis (1984), where it is shown that the distribution of the smallest eigenvalues plays a particularly important role. 2 in the context of the method of steepest descent. 4 THE PRECONDITIONED CONJUGATE GRADIENT METHOD 29 to reduce the problem of minimizing the functional to that of minimizing the related functional where E being some nonsingular matrix.
3. The A:th step in the method of steepest descent. 11) recursively. The iterations can thus be formulated as follows: where k = 0, 1,.... Initially we compute g° = Hx° — b. 13) can be implemented with only one multiplication by H per iteration. Two multiplications are required if g* is computed directly. Convergence Analysis We shall now show that the sequence x°, x1, ... 8), and we shall estimate the rate of convergence. To describe convergence quantitatively it is necessary to measure the error in any vector x that is to be considered as an approximation to x.
Differential equations: a dynamical systems approach 1 by J.H. Hubbard, B.H. West