119 lines
2.9 KiB
Plaintext
119 lines
2.9 KiB
Plaintext
1.9:
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(define (+ a b)
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(if (= a 0) b (inc (+ (dec a) b))))
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(+ 4 5)
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(if (= 4 0) 4 (inc (+ (dec 4) 5)))
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(inc (+ 3 5))
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(inc (inc (+ 2 5)))
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(inc (inc (inc (+ 1 5))))
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(inc (inc (inc (inc (+ 0 5)))))
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(inc (inc (inc (inc 5))))
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(inc (inc (inc 6)))
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(inc (inc 7))
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(inc 8)
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9
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Recursive
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(define (+ a b)
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(if (= a 0) b (+ (dec a) (inc b))))
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(+ 4 5)
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(+ 3 6)
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(+ 2 7)
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(+ 1 8)
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(+ 0 9)
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9
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Iterative
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1.10:
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(define (A x y)
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(cond ((= y 0) 0)
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((= x 0) (+ 2 y))
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((= y 1) 2)
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(else (A (- x 1)
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(A x (- y 1))))))
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(A 1 10)
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(A 0 (A 1 9))
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(A 0 (A 0 (A 1 8)))
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(A 0 (A 0 (A 0 (A 1 7))))
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(A 0 (A 0 (A 0 (A 0 (A 1 6)))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 6)))))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))
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(A 0 (A 0 (A 0 (A 0 (A 0 10))))))
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(A 0 (A 0 (A 0 (A 0 12)))))
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(A 0 (A 0 (A 0 14))))
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(A 0 (A 0 16)))
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(A 0 18)
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22
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(A 2 4) = 16
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(A 3 3) = 16
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1.34
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(define (f g) (g 2))
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(f f) gives error Error: call of non-procedure: 2. This is because it is expecting the
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second argument to be a procedure (implicitly from the fact that it applies it to 2).
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1.35
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Golden ratio: phi = (1 + sqrt(5))/2
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Let x' be a fixed point of x |-> 1 + 1/x. Then
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x' = 1 + 1/x' = (x' + 1)/x'
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x'^2 = x' + 1
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x'^2 - x' - 1 = 0
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Solutions: (1 +- sqrt(1 + 4)) / 2. So phi is one of the fixed points.
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2.64
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partial-tree parititions the list into a left tree, the node, and a right tree. This
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is done such that the node is of size 1 and the left and right are equal for an odd-sized
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list and left is 1 smaller than right for an even sized list. partial-tree is then called
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on the left and right sub trees. The base case for the recursion is when the left and right
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trees are both of size 0. In this case, partial tree returns a leaf node (i.e. with null
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left and right pointers).
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Each element of the list is visited once, and for each visit, make-tree is called (which is
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just a call to list, which is assumed to be O(1)). So list->tree is O(n).
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2.77
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It is necessary to call apply-generic for each layer of tagged data. Previously
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there were only generic functions defined for complex data tagged with 'rect or
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'polar. Adding the complex package means we call apply-generic once for the complex
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tag and then again for the 'rect or 'polar tag, using the same real-part, imag-part,
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magnitude and angle functions. This works because these are defined in terms of
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apply-generic, which in turn uses the package definition.
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For z as in the example,
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(magnitude z)
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(apply-generic 'magnitude z)
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(magnitude ('rectangular 3 . 4))
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(apply-generic 'magnitude ('rectangular 3 . 4))
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(sqrt (+ (square 3) (square 4)))
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5
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