## November 29, 2008

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Saturday, November 29, 2008
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SICP
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SICP Exerice 1.29

Simpson's Rule is a more accurate method of numerical integration than the method illustrated above. Using Simpson's Rule, the integral of a function f between a and b is approximated as

(h/3)(y0 + 4y1 + 2y2 + 4y3 + 2y4 + ...+ 2yn-2 + 4yn-4 + yn)

where h = (b - a)/n, for some even integer n, and yk = f(a + kh). (Increasing n increases the accuracy of the approximation.) Define a procedure that takes as arguments f, a, b, and n and returns the value of the integral, computed using Simpson's Rule. Use your procedure to integrate cube between 0 and 1 (with n = 100 and n = 1000), and compare the results to those of the integral procedure shown above.

(defun cube (x) (* x x x))

(defun sum (term a next b)

(if (> a b)

0

(+ (funcall term a)

(sum term (funcall next a) next b))))

(defun integral (f a b n)

(let ((h (/ (- b a) n)))

(labels ((yk (k)

(funcall f (+ a (* k h))))

(inc2 (n) (+ n 2)))

(* (/ h 3)

(+ (yk 0)

(* 4 (sum #'yk 1 #'inc2 (- n 1)))

(* 2 (sum #'yk 2 #'inc2 (- n 2)))

(yk n))))))

SICP Exercise 1.30

The sum procedure above generates a linear recursion. The procedure can be rewritten so that the sum is performed iteratively. Show how to do this by filling in the missing expressions in the following definition:

(define (sum term a next b)

(define (iter a result)

(if <??>

<??>

(iter <??> <??>)))

(iter <??> <??>))

(defun new-sum (term a next b)

(labels ((iter (a result)

(if (> a b)

result

(iter (funcall next a)

(+ result (funcall term a))))))

(iter a 0)))

(defun inc (x) (+ x 1))

SICP Exercise 1.31

a. The sum procedure is only the simplest of a vast number of similar abstractions that can be captured as higher-order procedures.51 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 approximations to using the formula

(pie/4)=((2/3)(4/3)(4/5)(6/5)(6/7)(8/7)...)

b. If your product procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

(defun product (term a next b)

(if (> a b)

1

(* (funcall term a)

(product term (funcall next a) next b))))

(defun new-product (term a next b)

(labels ((iter (a result)

(if (> a b)

result

(iter (funcall next a) (* result (funcall term a))))))

(iter a 1)))

(defun factorial (n)

(labels ((ident (x) x))

(product #'ident 1 #'inc n)))

(defun my-pie (n)

(labels ((num-term (n)

(if (oddp n)

(- n 1)

n))

(den-term (n)

(if (evenp n)

(- n 1)

n)))

(float(* 4

(/ (product #'num-term 3 #'inc n)

(product #'den-term 3 #'inc n))))))

SICP Exercise 1.32

a. Show that sum and product (exercise 1.31) 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)

Accumulate takes as arguments the same term and range specifications as sum and product, together with a combiner procedure (of two arguments) that specifies how the current term is to be combined with the accumulation of the preceding terms and a null-value that specifies what base value to use when the terms run out. Write accumulate and show how sum and product can both be defined as simple calls to accumulate.

b. If your accumulate procedure generates a recursive process,write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

(defun accumulate (combiner null-value term a next b)

(if (> a b)

null-value

(funcall combiner (funcall term a)

(accumulate combiner null-value term

(funcall next a)

next b))))

(defun new-accumulate (combiner null-value term a next b)

(labels ((iter (a result)

(if (> a b)

result

(iter (funcall next a)

(funcall combiner result (funcall term a))))))

(iter a null-value)))

SICP Exercise 1.33

You can obtain an even more general version of accumulate (exercise 1.32) by introducing the notion of a filter on the terms to be combined. That is, combine only those terms derived from values in the range that satisfy a specified condition. The resulting filtered-accumulate abstraction takes the same arguments as accumulate, together with an additional predicate of one argument that specifies the filter. Write filtered-accumulate as a procedure. Show how to express the following using filtered-accumulate:

a. the sum of the squares of the prime numbers in the interval a to b (assuming that you have a prime? predicate already written)

b. the product of all the positive integers less than n that arerelatively prime to n (i.e., all positive integers i < n suchthat GCD(i,n) = 1).

(defun accumulate-if (filter combiner null-value term a next b)

(cond ((> a b) null-value)

((funcall filter a)

(funcall combiner

(funcall term a)

(accumulate-if filter

combiner

null-value

term

(funcall next a)

next

b)))

(t (accumulate-if filter

combiner

null-value

term

(funcall next a)

next

b))))

(defun prime-squared-sum (a b)

(accumulate-if 'primep

'+

0

#'(lambda(x) (* x x))

a

'inc

b))

(defun rel-prime-product (n)

(labels ((rel-prime-p (x)

(if (= 1 (gcd x n))

t)))

(accumulate-if #'rel-prime-p

'*

1

#'(lambda(x) x)

1

'inc

(- n 1))))

SICP Exercise 1.35

Show that the golden ratio (section 1.2.2) is a fixed point of the transformation x - 1 + 1/x, and use this fact to compute GO by means of the fixed-point procedure.

(defconstant +tolerance+ .00001)

(defun fixed-point (f first-guess)

(labels ((close-enough-p (v1 v2)

(< (abs (- v1 v2)) +tolerance+))

(try (guess)

(let ((next (funcall f guess)))

(if (close-enough-p guess next)

next

(try next)))))

(try first-guess)))

SICP Exercise 1.36

Modify fixed-point so that it prints the sequence of approximations it generates, using the newline and display primitives shown in exercise 1.22. Then find a solution to xx = 1000 by finding a fixed point of x log(1000)/log(x). (Use Scheme's primitive log procedure, which computes natural logarithms.) Compare the number of steps this takes with and without average damping. (Note that you cannot start fixed-point with a guess of 1, as this would cause division by log(1) =

0.)

(defun fixed-point-print (f first-guess)

(labels ((close-enough-p (v1 v2)

(< (abs (- v1 v2)) +tolerance+))

(try (guess)

(print guess)

(let ((next (funcall f guess)))

(if (close-enough-p guess next)

next

(try next)))))

(try first-guess)))

SICP Exercise 1.37

a. An infinite continued fraction is an expression of the form

f = N1/(D1+(N2/(D2+(N3/(D3+...

As an example, one can show that the infinite continued fraction expansion with the Ni and the Di all equal to 1 produces 1/phi , where phi is the golden ratio (described in section 1.2.2). One way to approximate an infinite continued fraction is to truncate the expansion after a given number of terms. Such a truncation -- a so-called k-term finite continued fraction -- has the form

N1/(D1+(N2/(...+Nk/Dk)

Suppose that n and d are procedures of one argument (the term index i) that return the Ni and Di of the terms of the continued fraction. Define a procedure cont-frac such that evaluating (cont-frac n d k) computes the value of the k-term finite continued fraction. Check your procedure by approximating1/phi using

(cont-frac (lambda (i) 1.0)

(lambda (i) 1.0)

k)

for successive values of k. How large must you make k in order to get an approximation that is accurate to 4 decimal places?b. If your cont-frac procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

(defun cont-frac (n d k &optional (i 1))

(if (> i k)

0

(/ (funcall n i)

(+ (funcall d i) (cont-frac n d k (+ i 1))))))

(defun cont-frac-iter (n d k &optional (i k) (result 0))

(if (< i 1)

result

(cont-frac-iter n d k

(- i 1)

(/ (funcall n i)

(+ (funcall d i) result)))))

SICP Exercise 1.38

In 1737, the Swiss mathematician Leonhard Euler published a memoir De Fractionibus Continuis, which included a continued fraction expansion for e - 2, where e is the base of the natural logarithms. In this fraction, the Ni are all 1, and the Di are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, .... Write a program that uses your cont-frac procedure from exercise 1.37 to approximate e, based on Euler's expansion.

(defun d (i)

"The D function to pass to cont-frac"

(cond ((= i 2) 2)

((zerop (mod (+ 1 i) 3)) (- i (/ i 3)))

(t 1)))

SICP Exercise 1.39

A continued fraction representation of the tangent function was published in 1770 by the German mathematician J.H. Lambert:

tan r = r/(1-(r^2/(3-(r^2/5-....

where x is in radians. Define a procedure (tan-cf x k) that computes an approximation to the tangent function based on Lambert's formula. K specifies the number of terms to compute, as in exercise 1.37.

(defun tan-cf (x k)

(labels ((tan-n (i)

(if (= i 1)

x

(- (* x x))))

(tan-d (i)

(if (= i 1)

1

(- (* 2 i)

1))))

(cont-frac #'tan-n #'tan-d k)))

SICP Exercise 1.40

Define a procedure cubic that can be used together with the

newtons-method procedure in expressions of the form (newtons-method

(cubic a b c) 1) to approximate zeros of the cubic x3 + ax2 + bx + c.

;;; Scheme functions redefined in CL

(defconstant +tolerance+ 0.00001)

(defconstant +dx+ 0.00001)

(defun deriv (g)

(lambda (x)

(/ (- (funcall g (+ x +dx+)) (funcall g x))

+dx+)))

(defun newton-transform (g)

(lambda (x)

(- x (/ (funcall g x) (funcall (deriv g) x)))))

(defun newton-method (g guess)

(fixed-point (newton-transform g) guess))

;;; Cubic function

(defun cubic (a b c)

(lambda (x)

(+ (* x x x)

(* a x x)

(* b x)

c)))

SICP Exercise 1.41

Define a procedure double that takes a procedure of one argumentasargument and returns a procedure that applies the original proceduretwice. For example, if inc is a procedure that adds1 to its argument,then (double inc) should be a procedure that adds 2. What value is returned by (((double (double double)) inc) 5)

(defun double (procedure)

(lambda (x)

(funcall procedure (funcall procedure x))))

SICP Exercise 1.42

Let f and g be two one-argument functions. The composition f afterg is defined to be the function x f(g(x)). Define a procedure compose that implements composition. For example, if inc is a procedure that adds 1 to its argument,((compose square inc) 6) 49

(defun compose (f g)

(lambda (x)

(funcall f (funcall g x))))

SICP Exercise 1.43

If f is a numerical function and n is a positive integer, then we can form the nth repeated application of f, which is defined to be the function whose value at x is f(f(...(f(x))...)). For example, if f is the function x x + 1, then the nth repeated application of f is the function x x + n. If f is the operation of squaring a number, then the nth repeated application of f is the function that raises its argument to the 2nth power. Write a procedure that takes as inputs a procedure that computes f and a positive integer n and returns the procedure that computes the nth repeated application of f. Your procedure should be able to be used as follows:

((repeated square 2) 5)

625

(defun repeated (f n &optional (count 1))

(if (>= count n)

(lambda (x) (funcall f x))

(compose (repeated f n (+ count 1)) f)))

SICP Exercise 1.44

The idea of smoothing a function is an important concept in signal processing. If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f(x - dx), f(x), and f(x + dx). Write a procedure smooth that takes as input a procedure that computes f and returns a procedure that computes the smoothed f. It is sometimes valuable to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtained the n-fold smoothed function. Show how to generate the n-fold smoothed function of any given function using smooth and repeated from exercise 1.43.

(defun smoothed (f)

(lambda (x)

(/ (+ (funcall f (- x +dx+))

(funcall f x)

(funcall f (+ x +dx+)))

3)))

(defun n-fold-smoothed (f n)

(repeated (smoothed f) n))

SICP Exercise 1.45

We saw in section 1.3.3 that attempting to compute square roots by naively finding a fixed point of y x/y does not converge, and that this can be fixed by average damping. The same method works for finding cube roots as fixed points of the average-damped y x/y2. Unfortunately, the process does not work for fourth roots -- a single average damp is not enough to make a fixed-point search for y x/y3 converge. On the other hand, if we average damp twice (i.e., use the average damp of the average damp of y x/y3) the fixed-point search does converge. Do some experiments to determine how many average damps are required to compute nth roots as a fixed-point search based upon repeated average damping of y x/yn-1. Use this to implement a simple procedure for computing nth roots using fixedpoint, average-damp, and the repeated procedure of exercise 1.43. Assume that any arithmetic operations you need are available as primitives.

;;; Scheme functions redefined in CL

(defun average-damp (f)

(lambda (x) (/ (+ x (funcall f x))

2)))

;;; End scheme functions

(defun nth-root (x n times)

(fixed-point (repeated (average-damp (lambda (y)

(/ (+ y

(/ x (expt y (- n 1))))

2)))

times)

1.0))

SICP Exercise 1.46

Several of the numerical methods described in this chapter are instances of an extremely general computational strategy known as iterative improvement. Iterative improvement says that, to compute something, we start with an initial guess for the answer, test if the guess is good enough, and otherwise improve the guess and continue the process using the improved guess as the new guess. Write a procedure iterative-improve that takes two procedures as arguments: a method for telling whether a guess is good enough and a method for improving a guess. Iterative-improve should return as its value a procedure that takes a guess as argument and keeps improving the guess until it is good enough. Rewrite the sqrt procedure of section 1.1.7 and the fixed-point procedure of section 1.3.3 in terms of iterative-improve.

(defun iterative-improve (good-enough-p improve)

(lambda (x)

(labels ((next-guess (guess)

(let ((improved-guess (funcall improve guess)))

(if (funcall good-enough-p guess improved-guess)

improved-guess

(next-guess improved-guess)))))

(next-guess x))))

(defun sqrt-improve (x)

(funcall (iterative-improve (lambda (guess z)

(let ((ratio (/ guess z)))

(and (< ratio 1.001) (> ratio 0.999))))

(lambda (guess)

(/ (+ guess (/ x guess))

2)))

1.0))

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