SICP Exercise 1.29 – Create more accurate method of numerical integration using Simpson’s Rule.
The following process works.
(define (sympson f a b n) (define (term-num x) (if (even? x) 2 4) ) (define (y k) ( f (+ a (* k h)))) (define h (/ (- b a) n)) (define (next x) (+ x 1)) (define (term x) (* (term-num x) ( y x )) ) (* (/ h 3.0) (+ (y 0) (sum term 1 next (- n 1)) (y n))) ) [shell] Then let's compare this method to the one which isn't using Simpson's Rule. First we write the code as follows: [shell] (integral cube 0 1 0.001) (sympson cube 0 1 10) (sympson cube 0 1 100)
And we’ll get the following output.
0.24999987500000073 0.25 0.25
It looks like the new one outperforms the previous one.
SICP Exercise 1.30 – Rewrite sum method as a linear iterative process.
You can easily solve this by using what we’ve learned so far.
(define (sum term a next b) (define (sum-iter a result) (if (> a b) result (sum-iter (next a) (+ result (term a)))) ) (sum-iter a 0) )
SICP Exercise 1.31(a) – Write an analogous procedure called product that returns the product of the values of a function at points over a given range. Show how to define factorial in terms of product. Also use product to compute approximationas to PI using the formula: PI/4 = (2*4*4*6*6*8*…) / (3*3*5*5*7*7*…)
We can get the definition of product by changing sum method we’ve created just before.
(define (product term a next b) (define (product-iter a result) (if (> a b) result (product-iter (next a) (* result (term a)))) ) (product-iter a 1) )
Then the definition of factorial is:
(define (factorial n) (product (lambda (x) x) 1 (lambda (x) (+ x 1)) n) )
And the definition of the quarter of PI is:
(define (qrt-pi) (define (term x) (if (even? x) (/ (+ x 2.0) (+ x 1)) (/ (+ x 1.0) (+ x 2))) ) (define (next x) (+ x 1) ) (product term 1 next 30000) )
To confirm these codes, we write as follows:
; Factorial (factorial 7) ; 5040 ; PI (* 4.0 (qrt-pi)) ; 3.14.....
It looks like this works well.
SICP Exercise 1.32(a) – Show that sum and product are both special cases of a still more general notion called accumulate that combines a collection of terms, using some general accumulation function:
(accumulate combiner null-value term a next b)
So let’s create this by using linear iterative process. We can easily get the abstraction by watching two functions( sum, product ) and extracting the common points.
(define (accumulate combiner null-value term a next b) (define (iter a result) (if (> a b) result (iter (next a) (combiner result (term a)))) ) (iter a null-value) )
Then two functions will be as follows:
(define (sum term a next b) (accumulate + 0 term a next b) ) (define (product term a next b) (accumulate * 1 term a next b) )
Now let’s check their behavior.
(define (total term from to) (sum term from (lambda (x) (+ x 1)) to) ) (define (factorial n) (product (lambda (x) x) 1 (lambda (x) (+ x 1)) n) ) (total (lambda (x) x) 1 10) ; 55 (total (lambda (x) (* x x)) 1 5) ; 55 (factorial 7) ; 5040