`#include "MusimatTutorial.h"`

Go to the source code of this file.

## Functions | |

MusimatTutorialSection (B0126) | |

Integer | iterFib (Integer n) |

Static Void | para1 () |

Integer | recurFib (Integer n) |

Static Void | para2 () |

Integer iterFib | ( | Integer | n ) |

MusimatTutorialSection | ( | B0126 | ) |

Definition at line 2 of file B0126.cpp.

References para1(), and para2().

{ Print("*** B.1.26 Fibonacci Numbers ***"); /***************************************************************************** B.1.26 Fibonacci Numbers In the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 377, 610, 987, 1597, 2584, . . . each subsequent term is the sum of its two immediately preceding values. For example, 8 = 5 + 3. This series, invented by Leonardo Pisano Fibonacci (1170-1250), is the solution to a problem he posed in his book "Liber Abaci": "A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?" An iterative method of computing the Fibonacci sequence is shown below in function iterFib(). *****************************************************************************/ para1(); // Step into this function to continue the tutorial para2(); // Step into this function to continue the tutorial }

Static Void para1 | ( | ) |

Definition at line 40 of file B0126.cpp.

References iterFib().

{ /***************************************************************************** Executing the following For loop, *****************************************************************************/ For ( Integer i = 1; i < 10; i = i + 1 ) Print(iterFib(i)); }

Static Void para2 | ( | ) |

Definition at line 62 of file B0126.cpp.

References recurFib().

{ /***************************************************************************** The recursive technique has crisper expressive power than the iterative approach because we see the inner structure of the sequence directly in the method of its construction. However, it is com- putationally much more expensive, especially for large n, because we must call the recurFib() method twice at each step, whereas iterFib() performs only one addition and minor data shuffling at each step. Here is an example where Knuth's "goodness" criterion depends upon con- text. If efficiency is paramount, the iterative approach is preferred; recursion is preferable if expressive crispness is most important. The Fibonacci sequence becomes relevant musically when we examine the ratios of subsequent terms: 1 2 3 5 8 13 -,-,-,-,-,--, ... 1 1 2 3 5 8 The corresponding sequence of quotients is 1, 200, 1.500, 1.670, 1.600, 1.625, 1.619, 1.617, 1.618, . . . Thus we see that the ratio of adjacent Fibonacci numbers converges rapidly to the value of the golden mean, phi. The Greek letter phi is commonly used to stand for the golden mean. This number appears in a wide range of natural designs, including the arrangements of petals in flowers, seed clusters, and pine cones. Studied at least since Euclid wrote his Elements, the golden mean has appeared consciously and unconsciously as a central design element in countless musical works (see section 9.16.1). *****************************************************************************/ For (Integer i = 1; i < 10; i++) Print(recurFib(i)); /***************************************************************************** prints the same sequence. It is useful to watch recurFib() compute using a debugger because you can see how it recursively enters itself with ever-changing values. *****************************************************************************/ }

Generated on Fri Nov 26 2010 16:18:26 for Musimat Tutorial by 1.7.2