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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "--------------------------
------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 51 "n
orm1.mws   Norm functions for vectors and matrices" }}{PARA 0 "" 0 "" 
{TEXT -1 62 "---------------------------------------------------------
-----" }}{PARA 0 "" 0 "" {TEXT -1 59 "The idea here is that, if we wri
te the functions ourselves," }}{PARA 0 "" 0 "" {TEXT -1 42 "we might p
ossibly understand what they do." }}{PARA 0 "" 0 "" {TEXT -1 54 "Of co
urse, the norms are part of Maple::LinearAlgebra." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra)
:" }}{PARA 0 "" 0 "" {TEXT -1 17 "The vector 2-norm" }}{PARA 0 "" 0 "
" {TEXT -1 32 "norm2vt(u) = sqrt(sum (u[i])^2) " }}{PARA 0 "" 0 "" 
{TEXT -1 17 "t means temporary" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 125 "norm2vt:=proc(u) local d,i,s;\nd := Dimension(u);\ns := 0;\nf
or i from 1 by 1 to d do s := s + u[i]^2 end do;\nsqrt(s)\nend proc: \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Test vector V1" }{MPLTEXT 1 
0 0 "" }{TEXT -1 1 "\n" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "V1 := Vector([1,-2,3,-2,-7]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#V1G-%'RTABLEG6%\"(Gp5\"-%'MATRIXG6#7'7#\"\"\"7#!\"#7
#\"\"$F/7#!\"(&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "norm2vt(V1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#
n#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "A simpler versio
n of norm2v " }}{PARA 0 "" 0 "" {TEXT -1 45 "norm2v(u) = sqrt(<u,u>)  \+
 One-line definition" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "norm2v:=u->
sqrt(Transpose(u).u):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "no
rm2v(V1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#n#\"\"\"\"\"#" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 59 "We now look at the 1-norm and the infini
ty-norm for vectors" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "We first define the 1-norm" }}{PARA 0 "" 0 "" {TEXT -1 
22 "norm1v(u) = sum |u[i]|" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
121 "norm1v:=proc(u) local d,i,s;\nd := Dimension(u);\ns := 0;\nfor i \+
from 1 by 1 to d do s := s + abs(u[i]) end do;\ns\nend proc: " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "norm1v(Vector([1,-2,3,-2,-7]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Now we def
ine the infinity-norm" }}{PARA 0 "" 0 "" {TEXT -1 22 "normiv(u\} = max
 |u[i]|" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "normiv:=proc(u) local d
,i,s,si;\nd := Dimension(u);\ns := abs(u[1]);\nfor i from 2 by 1 to d \+
do\nsi := abs(u[i]); \nif (si > s) then s := si end if\nend do:\ns\nen
d proc:\n " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "normiv(Vector([1,-2,3,-2,-7]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"(" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 14 "Test matrix M1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "M1 := Matrix([[1,2],[-3,1],[1,-4]]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#M1G-%'RTABLEG6%\")[`u?-%'MATRIXG6#7%7$\"\"\"\"
\"#7$!\"$F.7$F.!\"%%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "M
atrix Frobenius norm" }}{PARA 0 "" 0 "" {TEXT -1 39 "normfm(M) = sum m
[i,j]^2  = Trace M^t M" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "normfm:= \+
m->Trace(Transpose(m).m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "normfm(Matrix(M1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Matrix 1-norm" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "norm1m(M) = max norm1v(column vectors of M)" }}{PARA 0 "
" 0 "" {TEXT -1 13 "(uses norm1v)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
197 "norm1m:=proc(m) local i,dc,si,s;\ndc := ColumnDimension(m);\ns :=
 norm1v(Column(m,1));\nfor i from 1 by 1 to dc do\n  si := norm1v(Colu
mn(m,i));\n  if (si > s) then s := si end if;\nend do;\n  s\nend proc:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
17 "We try an example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "M1;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'RTABLEG6%\")[`u?-%'MATRIXG6#7%7$\"\"\"\"\"#7$!\"$F,7$F,!\"%%'Mat
rixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "norm1m(M1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 "Maple has its own function MatrixNorm for this task" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "MatrixNorm(M1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Matrix infinity-norm" }}{PARA 0 "
" 0 "" {TEXT -1 40 "normim(M) = max normiv(row vectors of M)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "normim:=m->norm1m(Transpose(m)):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Th
e example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "normim(M1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The Maple version of this is
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "MatrixNorm(M1,infinity
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "Transpose(M1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6%\"'%o/'-%'MATRIXG6#7$7%\"\"\"!\"$F,7%\"\"#F,!\"%%'MatrixG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "norm1m(Transpose(M1));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "MatrixNorm(Transpose(M1),1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"&" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "normim(Transpose(M1));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 26 "[ And,  \+
again, a la Maple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "MatrixNorm(Transpose(M1),infinity);" }}
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