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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "L-16 MCS 563 Wednesday 29 \+
September 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "In th
is worksheet we illustrate the relationship between least squares and \+
the backward error." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "The examp
le we consider appeared first in exercise 3 of section 1.5 on page 23,
 which continued through example 3.3 (page 74), example 5.6 (page 150)
, and example 6.4 (page 185)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "intrinsic_p := (x-sqrt(2))^3*(x+sqrt(2));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "empirical_p := evalf(expand(intrinsic_p),5);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := unapply(empirical_p,
x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sols := solve(empiri
cal_p,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pseudo_zeros :
= evalf(sols,5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "residua
ls := map(abs,map(f,[pseudo_zeros]));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "Now we will set up the least squares problem:" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "d := degree(empi
rical_p);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a := Vector([s
eq(coeff(empirical_p,x,i),i=0..d)]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "z1 := pseudo_zeros[1];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "S := (d,z) -> Matrix(d+1,d,(i,j) -> piecewise(i=j,-z,
i=j+1,1,0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "S1 := S(4,z
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "q1 := LinearAlgebra[
LeastSquares](S1,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r1 \+
:= a - S1.q1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "LinearAlge
bra[Norm](r1,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "v1 := V
ector([seq(abs(z1)^i,i=0..d)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
135 "The vector v1 with consecutive powers of z1 is parallel to the re
sidual vector computed by least squares, as verified by the following:
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "r1/r1
[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "n1 := LinearAlgebra
[Norm](v1,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "delta[1] :
= abs(f(z1))/(n1*10^(-5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "In
stead of evaluating the polynomial, we can also take the residual of t
he least squares computation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "LinearAlgebra[Norm](r1,1)/(n1*10^(-5));" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 48 "which leads to a delta[1] of the same magnitude." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The second ro
ot is a bit more interesting, since it has a larger error:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "z2 := pseudo_zeros[2];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "v2 := Vector([seq(abs(z2)^i,i=0..d)
]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "n2 := LinearAlgebra[
Norm](v2,1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "delta[2] :=
 abs(f(z2))/(n2*10^(-5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Not
e the significance of the delta as validity parameter.  In order for h
aving a valid zero, delta should be less than one, or not much larger \+
than one." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "W
e can generalize for the cluster of three roots:" }{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "z3 := pseudo_zeros[3]: z4 \+
:= pseudo_zeros[4]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "clus
ter := (x-z2)*(x-z3)*(x-z4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "expanded_cluster := expand(cluster);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 49 "b := [seq(coeff(expanded_cluster,x,i),i=0..d-1)];" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S3b := Matrix([[op(b), 0]
,[0,op(b)]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "S3 := Line
arAlgebra[Transpose](S3b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "q3 := LinearAlgebra[LeastSquares](S3,a);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "r3 := a - S3.q3;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "delta[3] := LinearAlgebra[Norm](r3,1)/10^(-5);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "If we take a larger cluster radius
, as in the book, example 6.4:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 46 "cluster := expand((x-1.40)*(x-1.41)*(x-1.42)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "b := [seq(coeff(clust
er,x,i),i=0..d-1)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S3b \+
:= Matrix([[op(b), 0],[0,op(b)]]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "S3 := LinearAlgebra[Transpose](S3b):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "q3 := LinearAlgebra[LeastSquares](S
3,a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r3 := a - S3.q3;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "r3/r3[1];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "delta[3] := LinearAlgebra[Norm](r3,
1)/10^(-5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The magnitude of t
his delta corresponds to the number of the book, which has 3400." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This delta[3]
 measures the validity of the cluster polynomial as a factor of the gi
ven polynomial." }{MPLTEXT 1 0 0 "" }}}}{MARK "54" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }