{VERSION 4 0 "IBM INTEL LINUX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# ROBIN STURM-LIOUVI LLE EIGENVALUES BY NEWTON'S METHOD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# First let's see if Maple can solve the problem already." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "# Take both constants = 1 (\"omega\" and \" beta\" in notes)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( tan(x) = - x, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# Tha t's not very informative. Try the alternative equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve( x*cos(x) + sin(x) = 0, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# This is the one root we are NOT i nterested in." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# Let's write a generalized \+ Newton iteration function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "newton := (f,x) -> evalf( x - f(x)/D(f)(x) );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 56 "# Define the recommended function for the Ro bin problem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "robin := x \+ -> x*cos(x) + sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# \+ From the graph (p. 77 of notes) we know that the first root is near 2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# Let's try for one of the l arger roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "newton(robi n, 7*Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin , %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "# Now let's try th e function used for the graph, but not advised for Newton." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "stupidrobin := x-> tan(x) + x;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "newton(stupidrobin, 7*Pi/2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "newton(stupidrobin, 7*3 .15/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrob in, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "newton(stupidrob in, 4.6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidr obin, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidr obin, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "newton(robin, \+ 4.6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "newton(stupidrobin, 7*3.14/2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(stupidrobin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# Where SHOULD this h ave gone?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "newton(robin, \+ 10.494);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(%*2/Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "newton(robin, 8.182);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "newton(robin, %);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "evalf(%*2/Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "13 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }