{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 " \+ " }{TEXT 258 41 "Eigenvalues for Spheroidal Wave Functions" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 " \+ A Maple worksheet written by Hans Volkmer, January 4, 2001 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " \+ send comments to volkmer@uwm.edu" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "This worksheet computes the eigenvalues " }{XPPEDIT 18 0 "lambda[n]^m;" "6#)&%'lambdaG6#%\"nG%\"mG" }{TEXT -1 1 "(" } {XPPEDIT 18 0 "gamma^2;" "6#*$%&gammaG\"\"#" }{TEXT -1 62 ") of the sp heroidal differential equation using approximation " }}{PARA 0 "" 0 " " {TEXT -1 47 "by eigenvalues of d by d tridiagonal matrices." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We use the notation of Meixner-Sch aefke and the shorthand " }{XPPEDIT 18 0 "q = gamma^2;" "6#/%\"qG*$%&g ammaG\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The program lambda(m,n,q,d) computes the eigenvalue " }{XPPEDIT 18 0 "lam bda[n]^m;" "6#)&%'lambdaG6#%\"nG%\"mG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "q = gamma^2;" "6#/%\"qG*$%&gammaG\"\"#" }{TEXT -1 44 " where d h as to be chosen sufficiently large" }}{PARA 0 "" 0 "" {TEXT -1 1 "(" } {XPPEDIT 18 0 "2*d;" "6#*&\"\"#\"\"\"%\"dGF%" }{TEXT -1 110 " has to b e at least n-m). In any case the computed values are upper bounds for \+ the eigenvalues. They converge " }}{PARA 0 "" 0 "" {TEXT -1 55 "monoto nically to the eigenvalue as d tends to infinity." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 63 "Note: By redefining Digits the required accuracy c an be chosen." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Usage: Hit enter several times and then use lambda(m,n,q,d) as indicated." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "restart;with(linalg):Digits:=10;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "trideven:=proc(m,q,d) # yi elds d by d submatrix if n-m is even" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local A,i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "for i from 1 to d do A[ i,i]:=(m+2*i-2.0)*(m+2*i-1)-2*q*((m+2*i-2)*(m+2*i-1)-1+m^2)/(2*m+4*i-5 )/(2*m+4*i-1) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for i from 1 t o d-1 do A[i,i+1]:=-q*(2*m+2*i-1)*(2*m+2*i)/(2*m+4*i-1)/(2*m+4*i+ 1) od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 2 to d do " } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " A[i,i-1]:=-q*(2*i-3)*(2*i-2)/(2* m+4*i-7)/(2*m+4*i-5) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "tridodd:=proc(m,q,d) # the d by d submatrix if n-m is odd" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local A,i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(d,d,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "for i from 1 to d do A[i,i]:=(m+2*i-1.0)*(m+2*i)-2*q*((m +2*i-1)*(m+2*i)-1+m^2)/(2*m+4*i-3)/(2*m+4*i+1) od;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 91 "for i from 1 to d-1 do A[i,i+1]:=-q*(2*m+2*i)* (2*m+2*i+1)/(2*m+4*i+1)/(2*m+4*i+3) od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 2 to d do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " A[i,i-1]:=-q*(2*i-2)*(2*i-1)/(2*m+4*i-5)/(2*m+4*i-3) od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "lambda:=proc(m ,n,q,d) # Meixner-Schaefke lambda_m^n(q) , choose d sufficiently large " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "local l,L,A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "if type(n-m,even) then A:=trideven(m,q,d);l:=(n-m) /2+1;else A:=tridodd(m,q,d);l:=(n-m-1)/2+1; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L:=[eigenvalues(A)];L:=sort(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L[l];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "First example: We compare with tab le in Abramowitz and Stegun, page 762 (note that " }{XPPEDIT 18 0 "lam bda[m,n];" "6#&%'lambdaG6$%\"mG%\"nG" }{TEXT -1 1 "=" }{XPPEDIT 18 0 " lambda[n]^m+q;" "6#,&)&%'lambdaG6#%\"nG%\"mG\"\"\"%\"qGF*" }{TEXT -1 22 " ) and find agreement." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "m:=1;n:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for q fro m 0 to 16 do print(q,lambda(m,n,q,7)+q-m*(m+1)) od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Second example: Display monotone dependence of \+ the approximations on d; see example mentioned " }}{PARA 0 "" 0 "" {TEXT -1 15 "in Chapter SW ." }{MPLTEXT 1 0 1 " " }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "m:=2;n:=4;q:=10.0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for d from 2 to 7 do print(d ,lambda(m,n,q,d)) od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Third ex ample: Plot eigencurves." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "m:=2;n:=4;q:=10.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot (q->lambda(m,n,q,7),-5.0..5.0);" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{MARK "22" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }