想起了 CASIO-82ES 调牛顿解方程

无事，看 SICP，顺带敲过来。

抄录的内容

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（还是移到了文章末尾。见 #Examples: square roots by newton’s method。）

想起了高中在上海的时候，我们高考可以使用计算器（一般都用 CASIO fx-82es 或者 CASIO fx-991es plus ），我们班就有不少男同学很喜欢折腾计算器……我们会在网上找 Casio fx 82ES 刷机教程，然后来刷别人的计算器，弄出各种花式的爆屏方法，或者一些使用的扩展功能。

By the way，CASIO 的计算器不是每款都用自己专门设计的芯片，好几个型号可能用一个芯片，只不过有的机型的功能被限制了。有时候这种限制可以通过硬件突破：比如 fx-82es 早期出的 A 版，就可以直接拆开，用 2B 铅笔把电路板上两条线路连通，就立马变身 fx-991es plus，功能多了很多，可以计算矩阵（上海高中是要学行列式和矩阵初步的），可以计算三元一次方程组、一元三次方程，可以进行进制转化，可以进行虚数的运算和方程求解，等等。（不要问我记得为什么这么清楚，我可是玩了这么久的前辈呢！何况我手边就有一个 fx991es plus……）。有时候这种限制要通过各种奇怪的 hack 来让它进入特定的模式。

其中有一个就是 “牛顿解方程” 模式，名字很霸气（毕竟和牛顿扯上关系了）。牛顿解方程很好用，比如求一个方程（一元）的解，直接把方程输到计算器就成（对于我这种粗心大意的又懒的人，简直不能更好）。一直不知道为什么这个刷机模式为什么叫“牛顿解方程”，记得当时余意老师讲过一种迭代求根的方法，我又不用，早忘了，但我会按计算器嘛！

没有计算器的数学考试我可以去死，宁可不写。2 位以上的计算我都不太习惯思考，sin 30 45 60 就没搞清楚过，从来都是直接按计算器。那时按计算器还挺快，在 15 班时还一群人拼谁刷机快……¹后来刷机的方法学的越来越多，可以刷出 base-n 进制转换，求方程组，求积分和导数（高中不教这个所以也不怎么会用），矩阵向量统计什么的，还有一些很 fancy 的爆机方法，最牛逼的是 LU WangChen 的那个，可以让计算器瞬间无法开机，除非用起子打开壳子把电池抠出来放一会。如此强大，以至于每次考数学前都要去欺负 Peng Wei，“乖点不然把你计算器刷死……”²

后来高考前把计算器换成了 991，现在还在用，不用刷就有了所有 82ES b 版按百来个键才能刷出来的功能，省去了刷机的麻烦，也少个那个乐趣。

翻到了以前刷机看的文档：

以前的生活，还真是有趣啊。

Examples: square roots by newton’s method

Procedures, as introduced above, are much like ordinary mathematical functions. they specify a value that is determined by one or more parameters. but there is an important difference between mathematical functions and computer procedures. procedures must be effective.

As a case in point, consider the problem of computing square roots. we can define the square-root functions as

… (省略一些)

this describes a perfectly legitimate mathematical function. we could use it to recognize wether one number is the square root of another, or to derive facts about square roots in general. on the other hand, the definition does not describe a procedure. indeed, it tells us almost nothing about how to actually find the square root of a given number. it will not help matters to rephrase this definition in pseudo-lisp:

(define (sqrt x)
    (the y (and (>= y 0)
                (= (square y) x))))

this only begs the question. （关键是那个 the y 怎么来的撒）

the contrast between function and procedure is a reflection of the general distinction between describing properties of things and describing how to do things, or , as it is something referred to , the distinction between declarative (whit is) descriptions, whereas in computer science we are usually concerned with imperative (how to) descriptions.

how does one compute square roots? the most common way is to use newtons’s method of successive approximations, which says that whenever we have a guess y for the value of the square root of a number x, we can perform a simple manipulation to get a better guess (one closer to the actual square root) by averaging y with x/y. for examplw, we can compute the square root of 2 as follows. suppose our initial guess is 1:

continuing this process, we obtain better and better approximations to the square root.

now let’s formative the process in term of procedures. we start with a value for the radicand (the number whose square root we are trying to compute) and a value for the guess. if the guess is good enough for our purposes, we are done; if not, we must repeat the process with an improved guess. we write this basic strategy as a procedure:

(define (sqrt-iter guess x)
    (if (good-enough? guess x)
        guess
        (sqrt-iter (improve guess x) x)))

A guess is improved by averaging it with the quotient of the radicand and the old guess:

(define (improve guess x)
    (average guess (/ x guess)))

where

(define (average x y)
        (/ (+ x y) 2))

we also have to say what we mean by “good enough,” the following will do for illustration, but it is not really a very good test. the idea is to improve the answer until it is close enough so that its square differs from the radicand by less than a predetermined tolerance.

(define (good-enough? guess x)
    (< (abs (- (square guess) x)) 0.001))

finally, we need a way to get started. for instance, we can always guess that the square root of any number is 1.

(define (sqrt x)
    (sqrt-iter 1.0 x))

if we type these definitions to te interpreter, we can use sqrt just as we can use any procedure:

(sqrt 9)
3.00009155...

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