<a name="docA001">
2010-02-19-16-20 start
This file Power Mean Inequality Curve 
http://freeman2.com/powmean2.htm 
is written during Liu,Hsinhan study
the book
The Cauchy-Schwarz Master Class
J. Michael Steele ★★★★★
<a name="docA002">
ISBN 978-0-521-54677-5
chapter 8 The Ladder of Power Means.
At first, LiuHH create a fake curve
But not satisfied. 
Start code from 2010-02-18-14-07
done  code  at  2010-02-19-12-??

<a name="docA003">　pPow　program
Page control panel is next
[[
<a name="powMeanBgn"> 2010-02-19-12-53 
Graph area size, W:[300]  H:[300]
x min:[-5], x max:[5] ; y min:[0], y max:[10]  ; 
x/y min/max : ▣auto scale, □user scale Click button 
10^[1] ; ▣+/0 , □+/0/- ;  [5] digits Help1  
"random5#" fill [20] numbers;  □integer;  
<a name="docA004">
Box 1, xj sequence   
Box 2, pj sequence   
Box 3, output   
Box 4, drawDot   
t bgn [-5]  , t end [5] steps [30]  "dt"  "ds"
Click to RUN "PowerMean" "wipe" "random5#" control center
 { ∑[k=n,k=1] pkxkt }1/t
Box 5, dots t values   
]]
  
<a name="docA005">
Graph area size, W:[300]  H:[300]
Let user decide drawing board width
and height.

x min:[-5], x max:[5] ; y min:[0], y max:[10]  ; 
Let user decide x-axis two end values
and y-axis two end values.

<a name="docA006">
x/y min/max : ▣auto scale, □user scale
Let user decide use auto scale or
use user scale.
If choose auto scale, above boxes 
values are ignored.

"Click button " let user go to 
Click button "PowerMean" quickly

<a name="docA007">
10^[1]  ;
Random generated in the range [0,1]
If fill 1 to [1], program stretch
random number to [0,10] (10^1=10)
If fill 2 to [2], program stretch
random number to [0,100] (10^2=100)

<a name="docA008">　pPow　program
 ▣+/0 , □+/0/- ;
If choose ▣+/0 , output random positive
or zero, no negative. This is default.
If choose ▣+/0/- , output random positive
or zero or negative. In this file
xj sequence and pj sequence must be +/0
Negative number is not allowed.

<a name="docA009">
 [5] digits 
If fill 5 to [5], output to 1.2345
If fill 8 to [8], output to 32.456789

Help1  
Bring user to here.

<a name="docA010">
"random5#"
is execution button. generate random
numbers. output to box 1 for xj sequence 
and output to box 2 for pj sequence

fill [20] numbers;
If fill 20 to [20], xj sequence has 20
numbers, pj sequence has 20 numbers.

<a name="docA011">
 □integer;  
If check □integer, xj sequence is
integer only. (pj can not be integer)

Box 1, xj sequence  
Let user define his own xj sequence
 
<a name="docA012">
Box 2, pj sequence   
Let user define his own pj sequence
pj must sum to one. tolerance=1.e-8
if abs(sum_pj-1)>tolerance ERROR.

Box 3, output   
output curve (t,Mt) coordinates values.

<a name="docA013">　pPow　program
Box 4, drawDot   
Output user specified t value (t,Mt) pair.
If want to know Mt value for t=1.23456
t=1.23456 may not occur in "Box 3, output"
User input 1.23456 to box 5, program
send (t=1.23456, Mt) pair to box 4
and draw a dot at t=1.23456.

<a name="docA014">
t bgn [-5]  , t end [5]
Program draw (t,Mt) pair lines, start
at t="t bgn [-5]", end at t="t end [5]"
If "t end [5]" > "t bgn [-5]" autoscale work.
If "t end [-5]" < "t bgn [5]" userscale work.

<a name="docA015">
steps [30] 
If fill 30 to "steps [30]", program draw
30 short straight lines to build curve.
minimum value is 10.

"dt" delete "t bgn" and "t end"
"ds" delete "steps"

<a name="docA016">
Click to RUN "PowerMean"
powMeanf() execution button is 
"PowerMean"

 "wipe"
to delete drawing

<a name="docA017">
"random5#"
second random button.

 control center
goto control center quickly

<a name="docA018">　pPow　program
 { ∑[k=n,k=1] pkxkt }1/t
Equation for Mt and t not=0

Box 5, dots t values   
default t value be
-1 0 1 2 
Output to box four

<a name="docA019">
Drawing output has a top red dash
and a bottom red dash lines.
Top one is M+∞=maximum value of xk
k=1 to n
Bottom one is M-∞=minimum value of 
xk, k=1 to n

<a name="docA020">
This program may not be perfect.
But you have what I have.
Happy playing !!
Thank you for visiting Freeman's page.
2010-02-19-17-25 stop



<a name="docA021">
2010-02-19-17-52 start
powmean2.htm contain the following
external javascript files
jsgraph2.js
tutelink.js
rocsitee.js
jslist1e.js
<a name="docA022">
Where 
graph code is jsgraph2.js,
link  list is tutelink.js,
Javascript program  list jslist1e.js
Freeman site banner file rocsitee.js

<a name="docA023">　pPow　program
If you want the drawing program 
work must include jsgraph2.js
Location is
http://freeman2.com/jsgraph2.js

Other *.js files use same domain 
http://freeman2.com/filename.js

<a name="docA024">
jsgraph2.js is XYGraph v2.3 code 
written by Mr. J. Gebelein. You 
can find code at
http://www.structura.info/XYGraph/XYGraph.zip
2010-02-19-18-00 stop

<a name="docA025">
2010-02-22-18-47 start
"Update 2010-02-22" made small 
change.
Allow "10^" box fill with non-
integer. For example
"10^ [ 1.2345 ]"
will create random number
range in [0, 17.16]
Code change is
[[
//arg2=parseInt(arg2);
arg2=parseFloat(arg2); //9902210800 use parseFloat()
]]

<a name="docA026">
If you want to keep drawing.
Use keyboard "copy screen"
function. Paste full screen
image to paint.exe and save
to file.
2010-02-22-18-53 stop

<a name="docA027">　pPow　program
2010-02-23-15-55 start
"Update 2010-02-23" change as below.
If in your application probability
is equal and if you have a data set
of twelve elements.
You can not fill twelve '1' to
"Box 2, pj sequence", because
twelve '1' sum to 12 which is
not equal to 1, program stop work.
<a name="docA028">
In this case you can click check
box "eq.p" (equal probability), 
program auto fill twelve 1/12
(0.08333333333333333)
for you, avoid hand fill in.

Code change please search for
"99022315"
2010-02-23-16-00 stop

<a name="docA029">
2010-02-24-11-20 start
"Update 2010-02-24" main change is 
let precision digits number work
when "10^" box filled noninteger.

"Update 2010-02-22" made small 
change. Correct one point, miss
another point.
"Update 2010-02-24" key point is
at time stamp "9902232247" and
"9902240733". 
<a name="docA030">
If user specify "[5] digits" in
[[
10^[  ]  ; +/0 , +/0/- ; [5] digits
]]
then program output five digits
random number even if user assign
10^noninteger.
Code change please search for
"9902232"

<a name="docA031">　pPow　program
"Update 2010-02-24" second change 
is add "complex" checkbox. This
file powmean2.htm do not use
complex random number. But the
fillRand5() function has complex
random number capability. If user
click check "complex" checkbox,
Box 1, xj sequence fill with
complex random number. 

<a name="docA032">
User can get complex random number
here and use it in other web page 
or other application program.
2010-02-24-11-34 stop

<a name="docA033">
2010-02-25-11-48 start
"Update 2010-02-24" third change.

2010-02-25 morning added one line
[1 ] pk power. If not one, use steps=300+
allow user to explore:
if pk NOT sum to one, what happen? 
<a name="docA034">
If you set pk power other than one. 
For example, fill '0.9' . 
All probability coefficient change 
as following
  pk=pk0.9
Before change, pk must sum to 1.
<a name="docA035">
After change, pk not sum to 1.
Curve become non-smooth. Jump to
infinity at t=0. You need set greater 
step number, for example change from
30 to 300 to see abrupt jump.

<a name="docA036">　pPow　program
This explore arrangement come from
textbook Exercise 8.3.
If set pk power box to 0.
Any number zero th power is one.
All pk become one, that is to 
ignore pk and this choice is 
what Exercise 8.3 doing.

<a name="docA037">
Do not jump from 1 to 0 directly.
drawing board show nothing for 
pPow=0. Change pPow to 0.99 first
see the trend.

<a name="docA038">
pk power = 0. has a drawing at 
DrawEx0803. You can see DrawEx0803 
to get an idea what happen if 
ignore all pk.
2010-02-25-12-09 stop

<a name="docA039">
2010-03-02-09-30 start
"Update 2010-03-02" change as following
1. Add alert ("9903011048", "9903011052")
length (GM) compare length (AM) OK, require ∑pk=1 exactly.
or
length^0.731 compare length^1.0?　NO ! See broken curve below
   if change "pk power" box value to
   non-one, second alert line show up.
   ("0.731" varies, depend on pk value)

<a name="docA040">
2. PowerMeanA modify input. ("9903011945")
   PowerMeanB not modify input box

3. add "JensenA", "JensenB" click button.
   in function powMeanf() add Jensen code
   2010-03-01-17-28 start Jensen (9903011728 start)
   2010-03-01-19-23  done Jensen (9903011923  done)

<a name="docA041">　pPow　program
Detail as following

1.
This program powmean2.htm add control box
"pk power" allow user change probability
coefficients. If change, prob. coef. 
sum will not be one and cause trouble.
This trouble-output is designed on 
purpose, let user see what happen
if compare length^1.1 with length^1.0
If "pk power" value＞1.0, then ∑pk＜1.0
If "pk power" value＜1.0, then ∑pk＞1.0

<a name="docA042">　pPow　program
2. and 3. 'A' button run default setting.
'B' button run user setting. User must 
uncheck "auto scale", check "user scale"
Previous version, only one execution
button "PowerMean". Now total four
execution buttons
"PowerMeanA" modify input. ("9903011945")
"PowerMeanB" not modify input box
"JensenA" modify input. 
"JensenB" not modify input box
<a name="docA043">
The default control panel boxes setting
was for x-sequence in [5, 20] range.
Need one execution button "PowerMean".
2010-02-26-17-59 start Exercise 8.7
2010-03-01-21-25  done Exercise 8.7
During Exercise 8.7 work, LiuHH need
to know eqn.AZ044 curve shape.
Start modify powmean2.htm 
function powMeanf(). Added
JensenDraw=0; //9903011810
to guide Jensen output. 
<a name="docA044">
When Jensen click button is used,
the default control boxes values
are not suitable for Jensen. Then
in Jensen click button added 
[[
QEboxtBgn.value=0;
QEboxtEnd.value=1.5;
QEboxStep.value=300;
QEboxXmin.value=0
QEboxXmax.value=1.5
QEboxYmin.value=-2  //was -100
QEboxYmax.value=10  //was  100
QEchkbxcD2.checked=1
QEchkbxcD1.checked=0
]]
<a name="docA045">
before call powMeanf()
Ymin=-100 and Ymax=100 are for
x-sequence in [0,17]. LiuHH also
test x-sequence in [0,1] then click
"Jensen" button, graph auto set
Ymin=-100 and Ymax=100 , all three
curves are flat on x-axis. I have 
no control !! For x-sequence in 
[0,1] I need Ymin=-1 and Ymax=1.
Then write "JensenA" button and
"JensenB" button.
<a name="docA046">　pPow　program
"JensenA" button reset several boxes
 value, suitable for x-seq.=[0,17]
"JensenB" button do not reset any
 box value, use boxes value as is
and call powMeanf() directly.
This is why write "JensenA" and B
two buttons.

<a name="docA047">
If first run "JensenA" then second
run "PowerMean", all control boxes
value are for Jensen, not suitable
for PowerMean. Split "PowerMean"
too
"PowerMeanB" button do not reset any
 box value,
<a name="docA048">
"PowerMeanA" button reset the following
[[
QEboxtBgn.value=-5;
QEboxtEnd.value=5;
QEboxStep.value=30;
QEchkbxcD2.checked=0
QEchkbxcD1.checked=1
]]
before call powMeanf()
Please click the right button which
is suitable for your case.

<a name="docA049">
Why added Jensen code? 
Partial document is recorded at (local)
http://freeman2.com/tute0032.htm#ch08c115

<a name="docA050">　pPow　program
What is Jensen output curve key
point? Which point I should pay
attention to Jensen curve?
Please see Jensen-curve-below 
document. There tell you two
key points to observe. Not repeat
here.
2010-03-02-10-29 stop

2010-03-03-15-44 done spelling check


<a name="docA051">
2010-03-06-11-15 start
"Update 2010-03-06" made the
following change

<a name="docA052">
First:
2010-03-03-21-08 added
＜a href="#contralCenter"＞
Old version, click "contralCenter"
goto ＜a name="powMeanBgn"＞
This version, click "contralCenter"
goto ＜a href="#contralCenter"＞

<a name="docA053">
Second:
Add code to find slope value
//eqn.AZ064 (eqn.8.35) 
var dfdtat1= //eqn.AZ064 , 9903040358
    sum1-sum0*Math.log(sum0);

<a name="docA054">
Third:
draw tangent at t=1 on black curve
See code at "9903041110"
2010-03-06-11-23 stop

<a name="docA055">
2012-04-03-13-52 start
"Update 2012-04-03" corrected spelling 
error. Web page surface spelling error 
is easier to find out. Copy screen and 
do the spelling check. But /*code notes*/, 
and "quoted code string" do not show up 
on web page surface. Many spelling error 
buried in deep. 

<a name="docA056">
On 2012-03-09 LiuHH uploaded spelling 
check program to next URL
http://freeman2.com/dict000a.htm 
From 2012-03-26 to 2012-04-03 LiuHH 
write pickdata.htm to extract text 
from //code notes and from 
'quoted code string' dig them out and 
check spelling error. 
<a name="docA057">
LiuHH use 
http://freeman2.com/powmean2.htm 
do the test run. Now upload code notes
spelling checked powmean2.htm .
I need time to test/debug pickdata.htm
when it is ready, I will upload to 
http://freeman2.com/pickdata.htm 
Today 2012-04-03 above URL is not valid.
2012-04-03-14-07 stop