Applying the preceding economic concepts, we may then derive various economic- and financial models and principles. As above, the two usual areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital.
For an arbitrary positive-definite Q we say E is a generalized Euclidean metric or distance function. Thus, the directions are not specified beforehand, but rather are determined sequentially at each step of the iteration. At step k one evaluates the current negative gradient vector and adds to it a linear 9.
There are three primary advantages to this method of direction selection. First, unless the solution is attained in less than n steps, the gradient is always nonzero and linearly independent of all previous direction vectors.
If the solution is reached before n steps are taken, the gradient vanishes and the process terminates— it being unnecessary, in this case, to find additional directions. Second, a more important advantage of the conjugate gradient method is the especially simple formula that is used to determine the new direction vector.
This simplicity makes the method only slightly more complicated than steepest descent. Third, because the directions are based on the gradients, the process makes good uniform progress toward the solution at every step. This is in contrast to the situation for arbitrary sequences of conjugate directions in which progress may be slight until the final few steps.
Although for the pure quadratic problem uniform progress is of no great importance, it is important for generalizations to nonquadratic problems. In the algorithm the first step is identical to a steepest descent step; each succeeding step moves in a direction that is a linear combination of the current gradient and the preceding direction vector.
The attractive feature of the algorithm is the simple formulae, 19 and 20for updating the direction vector. The method is only slightly more complicated to implement than the method of steepest descent but converges in a finite number of steps.
It is easiest to prove this by simultaneously proving a number of other properties of the algorithm. The conjugate gradient algorithm 17 — 20 is a conjugate direction method. We first prove ab and c simultaneously by induction. The second term vanishes by the induction hypothesis on c.
This proves cwhich also proves that the method is a conjugate direction method. Part c is the equation that verifies that the method is a conjugate direction method. The basis of the viewpoint is part b of the Conjugate Gradient Theorem.
Each step of the method brings into consideration an additional power of Q times g 0.
It is this observation we exploit. Let us consider a new general approach for solving the quadratic minimization problem. Therefore, the problem posed of selecting the optimal P k is solved by the conjugate gradient procedure.
The power of the conjugate gradient method is that as it progresses it successively solves each of the optimal polynomial problems while updating only a small amount of information.February 5th, - Matlab Chapter 2 Solution Masteringphysics Solution Chapter Chapter 8 Solution Hibbeler Luenberger Chapter 7 Solution Income Tax Chapter Solution February 21st, - Income Tax Chapter Solution 6 Chapter 7 Solution Teacherweb Earthwear Chapter Solution Kieso Chapter 19 Solution Earthwear Chapter 5 Solution February 16th.
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