Re: new procedure for converting a new recursive polynomial set into matrices
- To: mathgroup at smc.vnet.net
- Subject: [mg70906] Re: new procedure for converting a new recursive polynomial set into matrices
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Wed, 1 Nov 2006 03:55:47 -0500 (EST)
- References: <ehkcmn$jn9$1@smc.vnet.net>
I found a good reference for going from a tridiagonal matric to a recursive polynomial: http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102708710 Joanne Dombrowski Tridiagonal matrix representations of cyclic selfadjoint operators. Source: Pacific J. Math. 114, no. 2 (1984), 325?334 It appears to work well in Mathematica as well! %I A000001 %S A000001 1, 1, -1, 0, -2, 1, -1, -3, 4, -1, -3, -6, 14, -7, 1, -14, -24, 72, -48, 12, -1, -109, -172, 586, -449, 143, -20, 1, -1403, -2103, 7718, -6375, 2296, -402, 33, -1, -29354, -42588, 163595, -141144, 54448, -10718, 1094, -54, 1, -996633, -1416535, 5597100, -4956116, 1990080, -418458, 47881, -2929, 88, -1, -54785461, -76870204, 309093440, -278042336, 114356068, -24994552, 3050819, -208922, 7768, -143, 1 %N A000001 Fibonacci central triadiagonal matrices as a triangular sequence from a recursive polynomial definition %C A000001 Matrices: {{1}}, {{1, -1}, {-1, 1}}, {{1, -1, 0}, {-1, 1, -1}, {0, -1, 2}}, {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 2, -1}, {0, 0, -1, 3}}, {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 2, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 5}}, {{1, -1, 0, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {0, -1, 2, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, {0, 0, 0, -1, 5, -1}, {0, 0, 0, 0, -1, 8}} The Dombrowski paper defines a recursive polynomial form from the tridiagonal matrices: p[1,x]=1,p[2,x]=(x-b[1])/a[1] p[n,x]=((x-b[n-1])*p[n-1,x]-a[n-2]*p[n-2,x])/a[n-1] As long as b[n-1]/a[n-1] and a[n-2]/a[n-1] behave well ( rationally or like Integers) this definition is a good recursive polynomial on a tridiagonal matrix. Here I use: a[n]=-1 and b[n]=Fibonacci[n] %D A000001 Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325?334 %F A000001 M(n,m)=If[ n == m, Fibonacci[n], If[n == m - 1 || n == m + 1, -1, 0]] %e A000001 Triangular sequence: {1}, {1, -1}, {0, -2, 1}, {-1, -3, 4, -1}, {-3, -6, 14, -7, 1}, {-14, -24, 72, -48, 12, -1}, {-109, -172, 586, -449, 143, -20, 1}, {-1403, -2103, 7718, -6375,2296, -402, 33, -1}, {-29354, -42588, 163595, -141144, 54448, -10718, 1094, -54, 1} %t A000001 T[n_, m_] := If[ n == m, Fibonacci[n], If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]; Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] %O A000001 1 %K A000001 ,nonn, %A A000001 Roger Bagula and Gary Adamson (rlbagula at sbcglobal.net), Oct 30 2006 RH RA 192.20.225.32 RU RI > > >