(define integers (stream-cons 1 (add-streams integers ones))) (define A (stream-cons 1 (scale-stream 2 A))) (define (mul-streams a b) (stream-cons (* (stream-first a) (stream-first b)) (mul-streams (stream-rest a) (stream-rest b)))) (define B (stream-cons 1 (mul-streams B integers))) ; What is A? ; What is B?
时间: 2024-03-11 19:45:09 浏览: 71
Based on the code provided, A is an infinite stream of powers of 2, starting with 1.
B is an infinite stream where the nth element is the product of the first n integers. This is achieved by defining a stream of integers (called "integers") starting with 1, and then defining B as a stream where each element is the product of the corresponding element in "integers" and the previous element in B. This multiplication is done using the "mul-streams" function which takes two streams and returns a stream of their pairwise products.
So in summary, A is a stream of powers of 2 starting with 1, and B is a stream of factorials starting with 1.
相关问题
(define g (stream-cons (list 1) (stream-map2 (lambda (x ls) (append ls (list x))) (stream-rest integers) g)))
这是一个 Scheme 中关于无穷流(Infinite Stream)的定义。其中 `stream-cons` 和 `stream-map2` 是 Scheme 中用于构建无穷流的两个函数。
该定义实际上定义了一个无穷流 `g`,其中第一个元素是 `(list 1)`,后续的元素是将整数流(从 `integers` 无穷流中取得)和 `g` 流进行 `stream-map2` 操作后得到的结果。具体来说,这个 `stream-map2` 操作是将整数流中的每个元素 `x` 与 `g` 中的所有元素 `ls` 进行操作,操作结果是将 `x` 添加到 `ls` 的末尾,得到一个新的列表。然后将这些新的列表按顺序放在一起,得到一个新的无穷流。
需要注意的是,由于 Scheme 中的惰性计算特性,我们并不需要对整个无穷流进行计算,而是可以根据需要只计算其中的部分元素。例如,可以用 `(stream-ref g 10)` 来取得 `g` 流的前 10 个元素。
Demonstrate that you can generate the series for ex as (define exp-series (stream-cons 1 (integrate-series-tail exp-series))) Explain the reasoning behind this definition. Show how to generate the series for sine and cosine, in a similar way, as a pair of mutually recursive definitions.
Sure, I can help with that!
First, let me explain the reasoning behind the definition of the `exp-series` stream. The series expansion for `e^x` is given by:
```
e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...
```
This can be rewritten as:
```
e^x = 1 + x*(1 + x/2*(1 + x/3*(1 + x/4*(...))))
```
We can use this recursive definition to generate the stream of coefficients for the series. The `stream-cons` function is used to define the stream, and the `integrate-series-tail` function is used to integrate the series to generate the next term. The `integrate-series-tail` function takes a stream and integrates it term-by-term. In this case, we integrate the `exp-series` stream to generate the next term in the series. The first term in the stream is 1, since `e^0 = 1`.
Here's the code to define the `exp-series` stream:
```
(define exp-series
(stream-cons 1 (integrate-series-tail exp-series)))
```
Now, let's move on to generating the series for sine and cosine. The series expansions for sine and cosine are given by:
```
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...
```
We can use similar recursive definitions to generate the streams of coefficients for these series. We'll define two streams, `sin-series` and `cos-series`, and we'll use mutual recursion to define them. The `sin-series` stream will start with `x`, since `sin(0) = 0`, and the `cos-series` stream will start with `1`, since `cos(0) = 1`. We'll use the same `integrate-series-tail` function to integrate the series and generate the next term.
Here's the code to define the `sin-series` and `cos-series` streams:
```
(define sin-series
(stream-cons 0 (integrate-series-tail cos-series)))
(define cos-series
(stream-cons 1 (integrate-series-tail (stream-map - sin-series))))
```
In the definition of `cos-series`, we use `stream-map` to negate the terms in `sin-series`, since the coefficient of `x` in the expansion for `cos(x)` is `0`. We can then use these streams to generate approximations of `sin(x)` and `cos(x)` for any value of `x`.
I hope that helps! Let me know if you have any other questions.
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