Calculation Method Reading notes

# # # written in front

Method of calculation in the eighth week of this hasty course, I have not felt learned anything, may be to lecture too little. Each time the last week or a few days to prepare a course, feel that there is something to stay, especially in the last year encountered some algorithm matrix pushed to, found that they have to the freshman learned things back to the teacher. While I have just finished the test and have some impressions, I would like to quickly record some of the experience.

Because I do not listen to the class, can only say some of their own think will be helpful to their own things, is the subject of some ideas, as for the right, ...

# # # Taylor unfold with Lagrange

In fact, the calculation method, is a number of methods, for the computer is to solve some fitting problems. For this problem, Lagrange gives a good way, using Taylor expansion can be used to expand the majority of functions into a polynomial form, and better yet, this degree of expansion can indicate the quality of the fit. Think of the function as a polynomial, which is the idea of Taylor's unfolding, and the basic idea of these numerical methods, which can be said that there is no such a way to do so without such an expansion.

In fact, many places in mathematics use this specific technique, such as the isomorphism of "group" to "permutation group" in discrete mathematics. Many of the problems in this course are to solve the problem of multi-fetching Lagrange expansion, and the higher order can get better fitting effect. This is a slippery question.

# # Smooth

How to achieve smoothness, for many of the methods of determination of integral (such as newton-cotes formula, Gauss Formula), is the use of Lagrange expansion to solve these coefficients. The higher order is used to accelerate the convergence, and fundamentally, it is also a number of the same-order infinite small-scale manufacturing greater order infinitesimal (Romberg integral method).

For smooth, can also directly consider some continuity, from the original function continuous to n-order derivative continuous, such as Hermite interpolation method, spline interpolation method, which are some of the use of N-order derivative of the smooth processing. In this semester I also chose the course of graphics, in which I used a lot of similar smooth processing, such as pixelated screen display, surface display, polygon chamfer and so on.

# # # Iterative method

Personal feeling, the iterative method is the center of Computer computing method, it is also possible that I had little contact with this idea, I feel that this kind of thought is worthy of reflection and promotion.

The iterative method can also be used to accelerate the convergence, even reasonable control of the error (forecast correction formula), or choose the step length. Using this method is a variety of iterations, approximation (Newton iteration, Jacobi iteration) and so on. This method is also a good way to control acceleration, simply, to make a linear change in the change can be accelerated (such as ultra-relaxation method).

This scheme is also a general solution to some complex situations, and now is often used in the solution of neural networks and so on, in this calculation is also often used in the same way as the super-relaxation method of acceleration, and even a method to calculate the value of the step should be obtained.

# # # Descending dimension, ascending dimension

The thought of descending dimension and ascending dimension is also very common in computational methods, so it is very difficult for the computer to solve the analytic solution of integral and differential, even for many cases. Using this idea, there are newton-cotes formula, Gauss formula (integration into the original function calculation). In this regard, Taylor's expansion is a process of separating dimensions, and iterative is also a dimensionality reduction (simplifying a method that requires a large amount of storage or computation to a linear calculation). For example, the catch-up method turns n^2 's matrix decomposition into 3n storage and 3n iterations.

# # Construction

The final mention of the construction of this matter, this I think is the essence of mathematics, but, mathematicians generally will not tell you how he thought of this construction, usually I am so structured you follow my thoughts like this, this proves it, but you think you can not think how to construct. Maybe that's the difference between applied science and science. The use of good constructs often simplifies the problem, and even creates and digs conditions (using the root method of the remainder).

Using the L0-LN function in the polynomial method with n interpolation, the method can get a good fit function form, and the same Newton interpolation is used in this way, and can even be reduced to iterative.

# # # Dynamic Planning

Dynamic programming can be considered as a core of the algorithm design course, many algorithms are designed by this idea, but because this method is so applicable, also cause I often think also can not use out. Due to my algorithm is not really, the afternoon has just been abused, I will pick up some of the relevant methods to briefly say this. For example, the Romberg method for integral, matrix triangular decomposition method to find the root of the equation, Newton interpolation of the micro-quotient calculation. Some sub-problems are used to construct higher-order parent problem, and the child problem is used to solve the parent problem.

Calculation Method Reading notes