#+BEGIN_HTML
---
title: 1.3 - Formulating Abstractions with Higher-Order Procedures
layout: org
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* Procedures as Arguments

  #+BEGIN_SRC scheme :tangle yes
    ;; ===================================================================
    ;; 1.3.1: Procedures as Arguments
    ;; ===================================================================

    (define (sum term a next b)
      (if (> a b)
          0
          (+ (term a)
             (sum term (next a) next b))))

    (define (inc n) (+ n 1))

    (define (cube n) (* n n n))

    (define (sum-cubes a b)
      (sum cube a inc b))

    (define (identity x) x)

    (define (sum-integers a b)
      (sum identity a inc b))

    (define (pi-sum a b)
      (define (pi-term x)
        (/ 1.0 (* x (+ x 2))))
      (define (pi-next x)
        (+ x 4))
      (sum pi-term a pi-next b))

    (define (integral f a b dx)
      (define (add-dx x) (+ x dx))
      (* (sum f (+ a (/ dx 2.0)) add-dx b)
         dx))

  #+END_SRC

** Exercise 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

   #+BEGIN_EXAMPLE
     h
     - (y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + ... + 2y_(n-2) + 4y_(n-1) + y_n)
     3
   #+END_EXAMPLE
   
   where h = (b - a)/n, for some even integer n, and y_k = 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.

   ----------------------------------------------------------------------

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 1.29
     ;; -------------------------------------------------------------------

     (define (simpson-integral f a b n)
       (define h (/ (- b a) n))
       (define (y k)
         (f (+ a (* k h))))
       (define (simpson-term x)
         (cond ((= x 0) (y x))
               ((= x n) (y x))
               ((even? x) (* 2 (y x)))
               ((odd? x) (* 4 (y x)))))
       (* (/ h 3) (sum simpson-term 0 inc n)))
   #+END_SRC
   
** 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:

   #+BEGIN_SRC scheme
     (define (sum term a next b)
       (define (iter a result)
         (if <??>
             <??>
             (iter <??> <??>)))
       (iter <??> <??>))
   #+END_SRC

   ---

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 1.30
     ;; -------------------------------------------------------------------

     (define (sum term a next b)
       (define (iter a result)
         (if (> a b)
             result
             (iter (next a) (+ (term a) result))))
       (iter a 0))

   #+END_SRC

** 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.(3)  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 [pi] using the formula(4)

      #+BEGIN_EXAMPLE
        pi   2 * 4 * 4 * 6 * 6 * 8 ...
        -- = -------------------------
         4   3 * 3 * 5 * 5 * 7 * 7 ...
      #+END_EXAMPLE
      
   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.

   ----------------------------------------------------------------------

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Example 1.31
     ;; -------------------------------------------------------------------

     (define (product-recursive term a next b)
       (if (> a b)
           1
           (* (term a)
              (product-recursive term (next a) next b))))

     (define (product-iter term a next b)
       (define (iter a result)
         (if (> a b)
             result
             (iter (next a) (* (term a) result))))
       (iter a 1))
   #+END_SRC
   
** Exercise 1.32
   a. Show that `sum' and `product' (*Note 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:

      #+BEGIN_SRC scheme
        (accumulate combiner null-value term a next b)
      #+END_SRC

      `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.

   ----------------------------------------------------------------------

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Example 1.32
     ;; -------------------------------------------------------------------

     (define (accumulate-recursive combiner null-value term a next b)
       (if (> a b)
           null-value
           (combiner (term a)
                     (accumulate-recursive combiner null-value term (next a) next b))))

     (define (accumulate-iter combiner null-value term a next b)
       (define (iter a result)
         (if (> a b)
             result
             (iter (next a) (combiner (term a) result))))
       (iter a null-value))
   #+END_SRC
   
** Exercise 1.33
   You can obtain an even more general version of
   `accumulate' (*Note 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 are
        relatively prime to n (i.e., all positive integers i < n such
        that GCD(i,n) = 1).

   ----------------------------------------------------------------------

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Example 1.33
     ;; -------------------------------------------------------------------

     (define (accumulate-filter predicate combiner null-value term a next b)
       (define (iter a result)
         (cond ((> a b) result)
               ((predicate a) (iter (next a) (combiner (term a) result)))
               (else (iter (next a) result))))
       (iter a null-value))

   #+END_SRC

* Constructing Procedures Using `Lambda'
** Exercise 1.34:
   Suppose we define the procedure

   #+BEGIN_SRC scheme
     (define (f g)
       (g 2))
   #+END_SRC
   
   Then we have

   #+BEGIN_SRC scheme
     (f square)
     4

     (f (lambda (z) (* z (+ z 1))))
     6
   #+END_SRC
   
   What happens if we (perversely) ask the interpreter to evaluate
   the combination `(f f)'?  Explain.

   ----------------------------------------------------------------------

   The call will fail, as ~(g 2)~ will evaluate to the form ~(2 2)~,
   which will fail to apply as ~2~ is a number, not a procedure.
   
   #+BEGIN_SRC scheme
     (f f)
     (f (f 2))
     (f (2 2))
     ;; The object 2 is not applicable.
   #+END_SRC

* Procedures as General Methods
  #+BEGIN_SRC scheme :tangle yes
    ;; -------------------------------------------------------------------
    ;; 1.3.3: Procedures as General Methods
    ;; -------------------------------------------------------------------

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

    (define (search f neg-point pos-point)
      (let ((midpoint (average neg-point pos-point)))
        (if (close-enough? neg-point pos-point)
            midpoint
            (let ((test-value (f midpoint)))
              (cond ((positive? test-value)
                     (search f neg-point midpoint))
                    ((negative? test-value)
                     (search f midpoint pos-point))
                    (else midpoint))))))

    (define (close-enough? x y)
      (< (abs (- x y)) 0.001))

    (define (half-interval-method f a b)
      (let ((a-value (f a))
            (b-value (f b)))
        (cond ((and (negative? a-value) (positive? b-value))
               (search f a b))
              ((and (negative? b-value) (positive? a-value))
               (search f b a))
              (else
               (error "Values are not of opposite sign" a b)))))

    (define tolerance 0.00001)

    (define (fixed-point f first-guess)
      (define (close-enough? v1 v2)
        (< (abs (- v1 v2)) tolerance))
      (define (try guess)
        (let ((next (f guess)))
          (if (close-enough? guess next)
              next
              (try next))))
      (try first-guess))

    (define (sqrt x)
      (fixed-point (lambda (y) (average y (/ x y)))
                   1.0))

  #+END_SRC
** Exercise 1.35:
   Show that the golden ratio [phi] (section *Note 1-2-2::) is a fixed
   point of the transformation x |-> 1 + 1/x, and use this fact to
   compute [phi] by means of the `fixed-point' procedure.

   ----------------------------------------------------------------------

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 1.35
     ;; -------------------------------------------------------------------

     (define phi
       (fixed-point (lambda (x) (+ 1 (/ 1 x)))
                    1.0))
   #+END_SRC
** Exercise 1.36:
   Modify `fixed-point' so that it prints the sequence of
   approximations it generates, using the `newline' and `display'
   primitives shown in *Note Exercise 1-22::.  Then find a solution to
   x^x = 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.)

   ----------------------------------------------------------------------

   #+BEGIN_SRC scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 1.36
     ;; -------------------------------------------------------------------

     (define (fixed-point-display f first-guess)
       (define (close-enough? v1 v2)
         (< (abs (- v1 v2)) tolerance))
       (define (try guess)
         (let ((next (f guess)))
           (display (list "Trying" next))
           (newline)
           (if (close-enough? guess next)
               next
               (try next))))
       (try first-guess))

     (fixed-point-display
      (lambda (x) (/ (log 1000) (log x)))
      1.5)

     ;(Trying 17.036620761802716)
     ;(Trying 2.436284152826871)
     ;(Trying 7.7573914048784065)
     ;(Trying 3.3718636013068974)
     ;(Trying 5.683217478018266)
     ;(Trying 3.97564638093712)
     ;(Trying 5.004940305230897)
     ;(Trying 4.2893976408423535)
     ;(Trying 4.743860707684508)
     ;(Trying 4.437003894526853)
     ;(Trying 4.6361416205906485)
     ;(Trying 4.503444951269147)
     ;(Trying 4.590350549476868)
     ;(Trying 4.532777517802648)
     ;(Trying 4.570631779772813)
     ;(Trying 4.545618222336422)
     ;(Trying 4.562092653795064)
     ;(Trying 4.551218723744055)
     ;(Trying 4.558385805707352)
     ;(Trying 4.553657479516671)
     ;(Trying 4.55677495241968)
     ;(Trying 4.554718702465183)
     ;(Trying 4.556074615314888)
     ;(Trying 4.555180352768613)
     ;(Trying 4.555770074687025)
     ;(Trying 4.555381152108018)
     ;(Trying 4.555637634081652)
     ;(Trying 4.555468486740348)
     ;(Trying 4.555580035270157)
     ;(Trying 4.555506470667713)
     ;(Trying 4.555554984963888)
     ;(Trying 4.5555229906097905)
     ;(Trying 4.555544090254035)
     ;(Trying 4.555530175417048)
     ;(Trying 4.555539351985717)
     ;;Value: 4.555539351985717
     ;
     (fixed-point-display
      (lambda (x) (average x (/ (log 1000) (log x))))
      1.5)

     ;(Trying 9.268310380901358)
     ;(Trying 6.185343522487719)
     ;(Trying 4.988133688461795)
     ;(Trying 4.643254620420954)
     ;(Trying 4.571101497091747)
     ;(Trying 4.5582061760763715)
     ;(Trying 4.555990975858476)
     ;(Trying 4.555613236666653)
     ;(Trying 4.555548906156018)
     ;(Trying 4.555537952796512)
     ;(Trying 4.555536087870658)
     ;;Value: 4.555536087870658

   #+END_SRC
** Exercise 1.37:
   a. An infinite "continued fraction" is an expression of the form

      #+BEGIN_EXAMPLE
                   N_1
        f = ---------------------
                       N_2
            D_1 + ---------------
                           N_3
                  D_2 + ---------
                        D_3 + ...
      #+END_EXAMPLE

      As an example, one can show that the infinite continued
      fraction expansion with the n_i and the D_i all equal to 1
      produces 1/[phi], where [phi] is the golden ratio (described
      in section *Note 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 finite continued fraction "k-term finite continued
      fraction"--has the form

      #+BEGIN_EXAMPLE
               N_1
        -----------------
                  N_2
        D_1 + -----------
              ...    N_K
                  + -----
                     D_K
      #+END_EXAMPLE

      Suppose that `n' and `d' are procedures of one argument (the
      term index i) that return the n_i and D_i 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
      approximating 1/[phi] using

      #+BEGIN_SRC scheme
        (cont-frac (lambda (i) 1.0)
                   (lambda (i) 1.0)
                   k)
      #+END_SRC

      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.

** 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 n_i are all 1, and the D_i are successively
   1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ....  Write a program that uses
   your `cont-frac' procedure from *Note Exercise 1-37:: to
   approximate e, based on Euler's expansion.

** Exercise 1.39:
   A continued fraction representation of the tangent function was
   published in 1770 by the German mathematician J.H. Lambert:

   #+BEGIN_EXAMPLE
                   x
     tan x = ---------------
                     x^2
             1 - -----------
                       x^2
                 3 - -------
                     5 - ...

   #+END_EXAMPLE

   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 *Note Exercise 1-37::.

* Procedures as Returned Values
  #+BEGIN_SRC scheme :tangle yes
    ;; -------------------------------------------------------------------
    ;; 1.3.4: Procedures as Returned Values
    ;; -------------------------------------------------------------------

    (define (average-damp f)
      (lambda (x) (average x (f x))))

    (define (sqrt x)
      (fixed-point (average-damp (lambda (y) (/ x y)))
                   1.0))

    (define (cube-root x)
      (fixed-point (average-damp (lambda (y) (/ x (square y))))
                   1.0))

    (define (deriv g)
      (lambda (x)
        (/ (- (g (+ x dx)) (g x))
           dx)))
    (define dx 0.00001)

    (define (cube x) (* x x x))

    (define (newton-transform g)
      (lambda (x)
        (- x (/ (g x) ((deriv g) x)))))

    (define (newtons-method g guess)
      (fixed-point (newton-transform g) guess))

    (define (sqrt x)
      (newtons-method (lambda (y) (- (square y) x))
                      1.0))

    (define (fixed-point-of-transform g transform guess)
      (fixed-point (transform g) guess))

    (define (sqrt x)
      (fixed-point-of-transform (lambda (y) (/ x y))
                                average-damp
                                1.0))

    (define (sqrt x)
      (fixed-point-of-transform (lambda (y) (- (square y) x))
                                newton-transform
                                1.0))


  #+END_SRC

** Exercise 1.40
   Define a procedure `cubic' that can be used together with the
   `newtons-method' procedure in expressions of the form
   
   #+begin_src scheme
        (newtons-method (cubic a b c) 1)
   #+end_src
   
   to approximate zeros of the cubic x^3 + ax^2 + bx + c.
   
** Exercise 1.41
   Define a procedure `double' that takes a procedure of one argument
   as argument and returns a procedure that applies the original
   procedure twice.  For example, if `inc' is a procedure that adds 1
   to its argument, then `(double inc)' should be a procedure that
   adds 2.  What value is returned by
   
   #+begin_src scheme
        (((double (double double)) inc) 5)
   #+end_src

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 1.41
     ;; -------------------------------------------------------------------

     (define (double f)
       (lambda (x) (f (f x))))

     (((double (double double)) inc) 5)
     ;Value: 21

   #+end_src
** Exercise 1.42
   Let f and g be two one-argument functions.  The "composition" f
   after g 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,
   
   #+begin_src scheme
        ((compose square inc) 6)
        49
   #+end_src

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 1.42
     ;; -------------------------------------------------------------------

     (define (compose f g)
       (lambda (x) (f (g x))))

   #+end_src
** 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 2^nth 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:
   
   #+begin_src scheme
        ((repeated square 2) 5)
        625
   #+end_src
   
   Hint: You may find it convenient to use `compose' from *Note
   Exercise 1-42::.

** 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 *Note
   Exercise 1-43::.
   
** Exercise 1.45
   We saw in section *Note 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/y^2.  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/y^3 converge.  On the other hand,
   if we average damp twice (i.e., use the average damp of the average
   damp of y |-> x/y^3) 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/y^(n-1).  Use this to implement a simple
   procedure for computing nth roots using `fixed-point',
   `average-damp', and the `repeated' procedure of *Note Exercise
   1-43::.  Assume that any arithmetic operations you need are
   available as primitives.
   
** 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 *Note 1-1-7:: and the `fixed-point' procedure
   of section *Note 1-3-3:: in terms of `iterative-improve'.