Writeup - ; M 2 = M*M = [(0, 1), (1, 1)]* [(0, 1), (1, 1)]...

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Stephanie Pitts “On my honor as a University of Colorado at Boulder student I have neither given nor received unauthorized assistance on this work.” 2000 naive: 0.179884910583496 rs: 0.000591039657592773 4000 naive: 0.4288330078125 rs: 0.000532865524291992 8000 naive: 1.24941492080688 rs: 0.000568866729736328 16000 naive: 3.82749605178833 rs: 0.000536918640136719 32000 naive: 13.5150260925293 rs: 0.000536918640136719 64000 naive: 53.0150129795074 rs: 0.000622034072875977 First 50: 44720160774243678502875116667917803275094537641593 Last 50: 52021580589777419718236165038410821903582102566626 Part B: (1) M N-1 = [(F N-3 , F N-2 ), (F N-2 , F N-1 )]; M N = [(F N-2 , F N-1 ),(F N-1, F N )]; M N+1 = [(F N-1 , F N ), (F N , F N+1 )]; (For N >= 1 ) This can be seen by simply multiplying the matrix M= [(0, 1), (1, 1)] by itself several times. The repeated squaring of this matrix gives the Fibonacci sequence in the second row and second column of the matrix. The repeated squaring is shown below. M = [(0, 1), (1, 1)]
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Unformatted text preview: ; M 2 = M*M = [(0, 1), (1, 1)]* [(0, 1), (1, 1)] = [(1, 1), (1, 2)] ; M 3 = M 2 *M= [(1, 1), (1, 2)]* [(0, 1), (1, 1)] = [(1, 2), (2, 3)] ; M 4 =M 3 *M= [(1, 2), (2, 3)]* [(0, 1), (1, 1)] = [(2, 3), (3, 5)] ; M 5 =M 4 *M= [(2,3), (3,5)]* [(0, 1), (1, 1)] = [(3, 5), (5, 8)]; As can be seen from the above multiplication my formula holds true. It must be this way because the Fibonacci sequence uses the last two numbers computed to get the next number (ie: F N =F N-2 + F N-1 ) . Thus, since the matrix we are multiplying has two ones in the second column the two numbers in the second row of the matrix it is being multiplied by must be the F N-2 and the F N-1 numbers, based on the conventions of matrix multiplication. The pattern that I found was based on repeated multiplication of the original matrix M until a pattern was determined. (2) G N = c N-1 (b) + c N-2 d(a)+ (N-2)c N-3 d(b)+(N-3)c N-4 d 2 (a)...
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Writeup - ; M 2 = M*M = [(0, 1), (1, 1)]* [(0, 1), (1, 1)]...

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