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SICP 05: final
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SICP/slide05.tm
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SICP/slide05.tm
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@ -102,8 +102,7 @@
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<tit|\<#7B97\>\<#6CD5\>\<#5BFC\>\<#8BBA\>\<#5BF9\><math|\<Theta\>>\<#8BB0\>\<#53F7\>\<#7684\>\<#5B9A\>\<#4E49\>>
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<\definition>
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For a given function <math|g<around*|(|n|)>>, we denote <samp|the set
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of funtions> by <math|\<Theta\><around*|(|g<around*|(|n|)>|)>>
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\<#5BF9\>\<#4E8E\>\<#7ED9\>\<#5B9A\>\<#7684\>\<#51FD\>\<#6570\><math|g<around*|(|n|)>>\<#FF0C\>\<#6211\>\<#4EEC\>\<#4F7F\>\<#7528\>\<#8BB0\>\<#6CD5\><math|\<Theta\><around*|(|g<around*|(|n|)>|)>>\<#8868\>\<#793A\>\<#4E00\>\<#7EC4\>\<#51FD\>\<#6570\>\<#7684\>\<#96C6\>\<#5408\>\<#FF1A\>
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<\equation*>
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\<Theta\><around*|(|g<around*|(|n|)>|)>=<around*|{|f<around*|(|n|)>:
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@ -112,8 +111,7 @@
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</definition>
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<\definition>
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For a given function <math|g<around*|(|n|)>>, we denote the set of
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funtions by <math|O<around*|(|g<around*|(|n|)>|)>>
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\<#5BF9\>\<#4E8E\>\<#7ED9\>\<#5B9A\>\<#7684\>\<#51FD\>\<#6570\><math|g<around*|(|n|)>>\<#FF0C\>\<#6211\>\<#4EEC\>\<#4F7F\>\<#7528\>\<#8BB0\>\<#6CD5\><math|O<around*|(|g<around*|(|n|)>|)>>\<#8868\>\<#793A\>\<#4E00\>\<#7EC4\>\<#51FD\>\<#6570\>\<#7684\>\<#96C6\>\<#5408\>\<#FF1A\>
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<\equation*>
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O<around*|(|g<around*|(|n|)>|)>=<around*|{|f<around*|(|n|)>:\<exists\>c,n<rsub|0>,\<forall\>n\<gtr\>n<rsub|0>,0\<leqslant\>f<around*|(|n|)>\<leqslant\>c*g<around*|(|n|)>|}>
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@ -121,8 +119,7 @@
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</definition>
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<\definition>
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For a given function <math|g(n)>, we denote the set of fuctions by
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<math|\<Omega\><around*|(|g<around*|(|n|)>|)>>
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\<#5BF9\>\<#4E8E\>\<#7ED9\>\<#5B9A\>\<#7684\>\<#51FD\>\<#6570\><math|g(n)>\<#FF0C\>\<#6211\>\<#4EEC\>\<#4F7F\>\<#7528\>\<#8BB0\>\<#6CD5\><math|\<Omega\><around*|(|g<around*|(|n|)>|)>>\<#8868\>\<#793A\>\<#4E00\>\<#7EC4\>\<#51FD\>\<#6570\>\<#7684\>\<#96C6\>\<#5408\>\<#FF1A\>
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<\equation*>
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\<Omega\><around*|(|g<around*|(|n|)>|)>=<around*|{|f<around*|(|n|)>:\<exists\>c,n<rsub|0>,\<forall\>n\<geqslant\>n<rsub|0>,0\<leqslant\>c*g<around*|(|n|)>\<leqslant\>f<around*|(|n|)>|}>
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@ -263,13 +260,127 @@
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</slide>
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<\slide>
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<tit|\<#6C42\>\<#5E42\>>
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<tit|\<#6C42\>\<#5E42\>\<#2014\>\<#2014\>\<#7EBF\>\<#6027\>\<#9012\>\<#5F52\>\<#7684\>\<#8BA1\>\<#7B97\>\<#8FC7\>\<#7A0B\>>
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<\equation*>
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b<rsup|n>=<choice|<tformat|<table|<row|<cell|b\<cdot\>b<rsup|n-1>>|<cell|,n\<gtr\>1>>|<row|<cell|b<rsup|0>>|<cell|,n=0>>>>>
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</equation*>
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<\session|scheme|default>
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<\unfolded-io|Scheme] >
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(define (expt-1 b n)
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\ \ (if (= n 0)
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\ \ \ \ \ \ 1
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\ \ \ \ \ \ (* b (expt-1 b (- n 1)))))
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<|unfolded-io>
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expt-1
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</unfolded-io>
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<\unfolded-io|Scheme] >
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(expt-1 1 0)
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<|unfolded-io>
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1
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</unfolded-io>
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<\unfolded-io|Scheme] >
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(expt-1 2 3)
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<|unfolded-io>
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8
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</unfolded-io>
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</session>
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<\big-table|<tabular|<tformat|<table|<row|<cell|\<#65F6\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>>|<cell|<math|\<Theta\><around*|(|n|)>>>>|<row|<cell|\<#7A7A\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>>|<cell|<math|\<Theta\><around*|(|n|)>>>>>>>>
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\<#590D\>\<#6742\>\<#5EA6\>
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</big-table>
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</slide>
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<\slide>
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<tit|\<#6C42\>\<#5E42\>\<#2014\>\<#2014\>\<#7EBF\>\<#6027\>\<#8FED\>\<#4EE3\>\<#7684\>\<#8BA1\>\<#7B97\>\<#8FC7\>\<#7A0B\>>
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\;
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<\equation*>
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b<rsup|n>=<choice|<tformat|<table|<row|<cell|b\<cdot\>b<rsup|n-1>>|<cell|,n\<gtr\>1>>|<row|<cell|b<rsup|0>>|<cell|,n=0>>>>>
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</equation*>
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<\session|scheme|default>
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<\unfolded-io|Scheme] >
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(define (expt-2 b n)
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\ \ (define (expt-iter b counter product)
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\ \ \ \ (if (= counter 0)
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\ \ \ \ \ \ \ \ product
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\ \ \ \ \ \ \ \ (expt-iter b
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (- counter 1)
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (* b product))))
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\ \
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\ \ (expt-iter b n 1))
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<|unfolded-io>
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expt-2
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</unfolded-io>
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<\unfolded-io|Scheme] >
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(expt-2 3 3)
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<|unfolded-io>
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27
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</unfolded-io>
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</session>
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<\big-table|<tabular|<tformat|<table|<row|<cell|\<#65F6\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>>|<cell|<math|\<Theta\><around*|(|n|)>>>>|<row|<cell|\<#7A7A\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>>|<cell|<math|\<Theta\><around*|(|1|)>>>>>>>>
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\<#590D\>\<#6742\>\<#5EA6\>
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</big-table>
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</slide>
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<\slide>
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<tit|\<#6C42\>\<#5E42\>\<#2014\>\<#2014\>\<#4F18\>\<#5316\>\<#65B9\>\<#6848\>>
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\;
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<\equation*>
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b<rsup|n>=<choice|<tformat|<table|<row|<cell|<around*|(|b<rsup|n/2>|)><rsup|2>>|<cell|,n\<#662F\>\<#5076\>\<#6570\>>>|<row|<cell|b\<cdot\>b<rsup|n-1>>|<cell|,n\<#662F\>\<#5947\>\<#6570\>>>>>>
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</equation*>
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<\session|scheme|default>
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<\unfolded-io|Scheme] >
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(define (fast-expt b n)
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\ \ (define (even? n)
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\ \ \ \ (= (remainder n 2) 0))
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\ \ (define (square x) (* x x))
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\ \
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\ \ (cond ((= n 0) 1)
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\ \ \ \ \ \ \ \ ((even? n) (square (fast-expt b (/ n 2))))
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\ \ \ \ \ \ \ \ (else (* b (fast-expt b (- n 1))))))
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<|unfolded-io>
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fast-expt
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</unfolded-io>
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<\unfolded-io|Scheme] >
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(fast-expt 3 3)
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<|unfolded-io>
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27
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</unfolded-io>
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</session>
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<\big-table|<tabular|<tformat|<table|<row|<cell|\<#65F6\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>>|<cell|<math|\<Theta\><around*|(|log<around*|(|n|)>|)>>>>|<row|<cell|\<#7A7A\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>>|<cell|<math|\<Theta\><around*|(|1|)>>>>>>>>
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\<#590D\>\<#6742\>\<#5EA6\>
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</big-table>
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</slide>
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<\slide>
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n|)>>\<#7684\>\<#8BC1\>\<#660E\>\<#3002\>
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</exercise>
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</slide>
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<\slide>
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<tit|\<#603B\>\<#7ED3\>>
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<\itemize>
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<item>\<#672F\>\<#8BED\>\<#56DE\>\<#987E\>\<#FF1A\>\<#6B63\>\<#5219\>\<#662F\>\<#4E2A\>\<#4EC0\>\<#4E48\>\<#9B3C\>
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<item>\<#65F6\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>\<#548C\>\<#7A7A\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>\<#7684\>\<#8BB0\>\<#53F7\>
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<item>\<#56DE\>\<#987E\>\<#5728\>\<#7B2C\>\<#4E09\>\<#8BFE\>\<#63D0\>\<#5230\>\<#7684\><scm|range>\<#51FD\>\<#6570\>
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<item>\<#6C42\><math|b<rsup|n>>\<#7684\>\<#4E0D\>\<#540C\>\<#5B9E\>\<#73B0\>\<#7684\>\<#4E0D\>\<#540C\>\<#65F6\>\<#7A7A\>\<#590D\>\<#6742\>\<#5EA6\>
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<item>\<#6709\>\<#4E00\>\<#4E9B\>\<#7B97\>\<#6CD5\>\<#7684\>\<#65F6\>\<#95F4\>\<#590D\>\<#6742\>\<#5EA6\>\<#5E76\>\<#4E0D\>\<#662F\>\<#663E\>\<#800C\>\<#6613\>\<#89C1\>\<#7684\>\<#FF0C\>\<#9700\>\<#8981\>\<#6570\>\<#5B66\>\<#53BB\>\<#8BC1\>\<#660E\>
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</itemize>
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\;
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\<#6590\>\<#6CE2\>\<#90A3\>\<#5951\>\<#6570\>\<#5217\>\<#7B2C\>n\<#9879\>\<#7684\>\<#8BA1\>\<#7B97\>\<#662F\>\<#4E00\>\<#4E2A\>\<#6811\>\<#5F62\>\<#9012\>\<#5F52\>\<#FF0C\>\<#5E76\>\<#4E0D\>\<#80FD\>\<#5F88\>\<#65B9\>\<#4FBF\>\<#5730\>\<#8F6C\>\<#6362\>\<#4E3A\>\<#7EBF\>\<#6027\>\<#8FED\>\<#4EE3\>\<#7684\>\<#8BA1\>\<#7B97\>\<#8FC7\>\<#7A0B\>\<#FF1A\>
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<\eqnarray*>
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<tformat|<table|<row|<cell|<with|mode|text|Fib><around*|(|n|)>>|<cell|=>|<cell|<around*|(|\<varphi\><rsup|n>-\<psi\><rsup|n>|)><around*|/|<sqrt|5>|\<nobracket\>>>>|<row|<cell|>|<cell|\<approx\>>|<cell|\<varphi\><rsup|n><around*|/|<sqrt|5>|\<nobracket\>>>>>>
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</eqnarray*>
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<\exercise>
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\<#56DE\>\<#987E\>\<#8BA1\>\<#7B97\>\<#6590\>\<#6CE2\>\<#90A3\>\<#5951\>\<#6570\>\<#5217\>\<#7B2C\>n\<#9879\>\<#7684\>\<#4E0D\>\<#540C\>\<#65B9\>\<#6CD5\>\<#7684\>\<#65F6\>\<#7A7A\>\<#590D\>\<#6742\>\<#5EA6\>\<#3002\>
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</exercise>
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<\render-exercise|\<#601D\>\<#8003\>\<#9898\>>
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\<#5728\>\<#6709\>\<#9650\>\<#7684\>\<#65F6\>\<#95F4\>\<#5185\>\<#FF0C\>\<#6BD4\>\<#5982\>1\<#5206\>\<#949F\>\<#FF0C\>\<#5F97\>\<#5230\>\<#80FD\>\<#591F\>\<#8BA1\>\<#7B97\>\<#51FA\>\<#6765\>\<#7684\>\<#6590\>\<#6CE2\>\<#90A3\>\<#5951\>\<#6570\>\<#5217\>\<#6700\>\<#5927\>\<#4E00\>\<#9879\>\<#3002\>
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</render-exercise>
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</slide>
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</slideshow>
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</body>
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<associate|par-columns|1>
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<associate|preamble|false>
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</collection>
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</initial>
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</initial>
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<\references>
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<\collection>
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<associate|auto-1|<tuple|1|?>>
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<associate|auto-2|<tuple|2|?>>
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<associate|auto-3|<tuple|3|?>>
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</collection>
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</references>
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