[Jprogramming] Matrix equation
Henry Rich
HenryHRich at nc.rr.com
Wed Aug 20 23:30:23 UTC 2014
This might be a good use of dissect, to see the shapes and values all at
once.
http://www.jsoftware.com/jwiki/Vocabulary/Dissect
Henry Rich
On 8/20/2014 7:27 PM, Mike Day wrote:
> Raul's right of course.
>
> Although Wx is indeed a scalar, Nc and so NcI are 1x1 matrices.
> NcI acts like a scalar, but if we replace NcI by scalar nci (say), then
> CT mp NcI mp C
> may be replaced by CT mp nci * C , or CT mp C * nci , etc.
>
> Mike
>
> On 20/08/2014 15:42, Mike Day wrote:
>> 0. Your J expression for Wx looks wrong to me.
>> If you want to get
>>
>> Wx = (1-(q0^2)-(q1^2)-(q2^2)-(q3^2))%2 in conventional arithmetic
>> notation
>> then you need
>> Wx=:(1-(q0^2)+(q1^2)+(q2^2)+(q3^2))%2
>> or
>> Wx=:(1-+/*:q)%2 NB. Your J-expression is effectively (1- -/
>> *:q)%2
>> or
>> [Wx=: -: -. +/*:,C NB. according to your definition of C
>>
>> 0
>> Maybe q = (q0, q1, q2, q3) is not always so simple.
>> Whatever its numerical value, Wx is scalar, and might perhaps be
>> regarded as 0.5 * (1 - C mp CT) .
>>
>>
>> 1. Nc = C.Nb.CT is scalar, and therefore NcI is also scalar.
>>
>>
>> 2. You don't state the vector l, so I can only use your stated value
>> for W, and I've used your stated value for B;
>> both of these were given to 3 places of decimals.
>>
>>
>> 3. A suitable J expression allowing for NbI as an overall factor,
>> since it's a left multiplier in both parts of
>>
>> (NbI-NbI*CT*NcI*C*NbI)*W + (NbI*CT*NcI*Wx) (conventional notation)
>>
>> appears to be
>>
>> dX=:NbI mp((I-CT mp NcI mp C mp NbI)mp W)+CT mp NcI mp Wx NB. J
>> notation
>>
>> where
>> I =: =i.#NbI NB. identity matrix
>>
>> Using my value of 0 for Wx:
>>
>> 25j20": dX=:NbI mp((I -CT mp NcI mp C mp NbI)mp W)+CT mp NcI mp Wx
>>
>> _0.33362737497592021000
>>
>> _0.21892578452135747000
>>
>> _0.10546486185920967000
>>
>> _0.24545356011509956000
>>
>> _0.10446141957003618000
>>
>>
>> Using your value of 0.5 for Wx:
>>
>> 25j20": dX=:NbI mp((I -CT mp NcI mp C mp NbI)mp W)+CT mp NcI mp %2
>>
>> _0.33362725373747026000
>>
>> _0.21892575956391414000
>>
>> _0.10546489384744806000
>>
>> _0.24545356063333446000
>>
>> _0.10446153412014780000
>>
>>
>> Evidently CT mp NcI is numerically small compared to the expression
>> involving W.
>>
>>
>> Please note that scalar part-expressions could be moved around, and
>> they may be applied with a simple * verb.
>> I haven't done that here.
>>
>>
>> Mike
>>
>> On 20/08/2014 08:43, "Papp Erik Tamás" wrote:
>>> Dear Forum,
>>>
>>> Can somebody help me to solve the folloving matrix equation?
>>>
>>> pps=:9!:11 NB. set print precision
>>> pps 20
>>> mp=: +/ . * NB. Matrix product
>>> NB.=======================================================================================
>>>
>>> NB. The following matrix equation have to be solve
>>> NB. dX=(NbI-NbI*CT*NcI*C*NbI)*W+NbI*CT*NcI*Wx
>>> NB.=======================================================================================
>>>
>>> NB. BT = B transpose
>>> BT=:|:B
>>> NB. Nb = BT*B
>>> Nb=:BT mp B
>>> NB. NbI = Nb inverse
>>> NbI=:%.Nb
>>> NB. W = BT * l
>>> W=: BT mp l
>>> NB. C = [q0 q1 q2 q3 0]
>>> C=:1 5$q0,q1,q2,q3,0
>>> NB. CT = C transpose
>>> CT=:|:C
>>> NB. Wx = (1-(q0^2)-(q1^2)-(q2^2)-(q3^2))%2
>>> Wx=:(1-(q0^2)-(q1^2)-(q2^2)-(q3^2))%2
>>> NB. Nc = C * Nb * CT
>>> Nc=:C mp Nb mp CT
>>> NB. NcI = Nc inverse
>>> NcI=:%.Nc
>>> NB.=======================================================================================
>>>
>>> ]Nc=:C mp Nb mp CT
>>> 5736.3999999999996
>>> ]Nc=:(C mp Nb) mp CT
>>> 5736.3999999999996
>>> ]Nc=:C mp (Nb mp CT)
>>> 5736.3999999999996
>>> NB.=======================================================================================
>>>
>>> 12j3":B
>>> 4.700 4.700 20.700 4.700 _3.300
>>> _4.700 _20.700 _11.300 _27.300 _8.000
>>> 16.000 6.600 22.600 0.000 8.000
>>> 2.200 12.200 8.200 _7.800 4.200
>>> _2.200 _18.200 6.200 _9.800 2.000
>>> 6.000 1.600 _2.400 10.000 8.000
>>> 9.200 _0.800 15.200 19.200 1.200
>>> _9.200 _5.200 _6.800 _2.800 _8.000
>>> 6.000 _12.400 3.600 _10.000 _2.000
>>> _10.300 _0.300 _24.300 _20.300 1.700
>>> 10.300 14.300 13.700 17.700 12.000
>>> _14.000 6.600 _17.400 10.000 _2.000
>>> _5.800 _15.800 _19.800 4.200 _3.800
>>> 5.800 29.800 _1.800 22.200 2.000
>>> _14.000 _2.400 _6.400 _10.000 _12.000
>>> $B
>>> 15 5
>>> 12j3":BT
>>> 4.700 _4.700 16.000 2.200 _2.200 6.000
>>> 9.200 _9.200 6.000 _10.300 10.300 _14.000
>>> _5.800 5.800 _14.000
>>> 4.700 _20.700 6.600 12.200 _18.200 1.600
>>> _0.800 _5.200 _12.400 _0.300 14.300 6.600
>>> _15.800 29.800 _2.400
>>> 20.700 _11.300 22.600 8.200 6.200 _2.400
>>> 15.200 _6.800 3.600 _24.300 13.700 _17.400
>>> _19.800 _1.800 _6.400
>>> 4.700 _27.300 0.000 _7.800 _9.800 10.000
>>> 19.200 _2.800 _10.000 _20.300 17.700 10.000
>>> 4.200 22.200 _10.000
>>> _3.300 _8.000 8.000 4.200 2.000 8.000
>>> 1.200 _8.000 _2.000 1.700 12.000 _2.000
>>> _3.800 2.000 _12.000
>>> $BT
>>> 5 15
>>> 12j3":Nb
>>> 1222.600 623.600 1555.000 853.000 611.300
>>> 623.600 2549.800 805.200 1754.000 602.300
>>> 1555.000 805.200 2985.800 1034.600 602.300
>>> 853.000 1754.000 1034.600 2936.600 602.300
>>> 611.300 602.300 602.300 602.300 607.300
>>> $Nb
>>> 5 5
>>> 12j6":NbI
>>> 0.004533 0.000333 _0.001772 _0.000311 _0.002826
>>> 0.000333 0.000749 _0.000128 _0.000382 _0.000573
>>> _0.001772 _0.000128 0.001127 0.000039 0.000754
>>> _0.000311 _0.000382 0.000039 0.000642 0.000016
>>> _0.002826 _0.000573 0.000754 0.000016 0.004295
>>> $NbI
>>> 5 5
>>> 12j3":W
>>> _981.642
>>> _1344.633
>>> _1326.828
>>> _1561.409
>>> _610.603
>>> $W
>>> 5 1
>>> 12j3":C
>>> 0.500 0.500 0.500 0.500 0.000
>>> $C
>>> 1 5
>>> 12j3":CT
>>> 0.500
>>> 0.500
>>> 0.500
>>> 0.500
>>> 0.000
>>> $CT
>>> 5 1
>>> 12j3":Wx
>>> 0.500
>>> $Wx
>>> 1
>>> 12j3":Nc
>>> 5736.400
>>> $Nc
>>> 1 1
>>> NcI
>>> 0.00017432536085349698
>>> $NcI
>>> 1 1
>>> NB.=======================================================================================
>>>
>>> NB. dX=(NbI-NbI*CT*NcI*C*NbI)*W+NbI*CT*NcI*Wx
>>> NB. dX=(NbI-NbI mp CT mp NcI mp C mp NbI) mp W+NbI mp CT mp NcI
>>> mp Wx
>>> NB.=======================================================================================
>>>
>>> NB. solution step by step
>>> dX=:(NbI-NbI mp CT mp NcI mp C mp NbI) mp W+NbI mp CT mp NcI mp Wx
>>
>>> NB. dX=f+i
>>> ]a=:NbI mp CT
>>> 0.0013909444659834052
>>> 0.00028633175856724745
>>> _0.00036699466085721166
>>> _5.9456044547029092e_6
>>> _0.0013142105194193007
>>> ]b=:a mp NcI
>>> 2.4247689595973174e_7
>>> 4.991488713605179e_8
>>> _6.3976476685240161e_8
>>> _1.0364696420582438e_9
>>> _2.291002230352313e_7
>>> ]c=:b mp C
>>> 1.2123844797986587e_7 1.2123844797986587e_7
>>> 1.2123844797986587e_7 1.2123844797986587e_7 0
>>> 2.4957443568025895e_8 2.4957443568025895e_8
>>> 2.4957443568025895e_8 2.4957443568025895e_8 0
>>> _3.1988238342620081e_8 _3.1988238342620081e_8
>>> _3.1988238342620081e_8 _3.1988238342620081e_8 0
>>> _5.1823482102912192e_10 _5.1823482102912192e_10
>>> _5.1823482102912192e_10 _5.1823482102912192e_10 0
>>> _1.1455011151761565e_7 _1.1455011151761565e_7
>>> _1.1455011151761565e_7 _1.1455011151761565e_7 0
>>> ]d=:c mp NbI
>>> 3.3727189656402491e_10 6.942883603207513e_11
>>> _8.8987726198450762e_11 _1.441671712780783e_12 _3.1866568738641721e_10
>>> 6.9428836032077935e_11 1.4292217412350897e_11
>>> _1.8318497076221252e_11 _2.9677417531211663e_13 _6.5598669749826067e_11
>>> _8.8987726198451706e_11 _1.8318497076220693e_11
>>> 2.3479025363938921e_11 3.8037882477597397e_13 8.4078558655125846e_11
>>> _1.4416717127807238e_12 _2.9677417531209235e_13
>>> 3.8037882477595918e_13 6.1624385209859802e_15 1.3621393066516946e_12
>>> _3.1866568738642073e_10 _6.5598669749824141e_11
>>> 8.4078558655125924e_11 1.3621393066517597e_12 3.0108592311420753e_10
>>> ]e=:NbI - d
>>> 0.0045328344164105271 0.00033293998348083668
>>> _0.0017724766597981138 _0.00031140912439777367 _0.0028261600905336174
>>> 0.00033293998348086812 0.00074882329681384913
>>> _0.00012756882532661765 _0.00038153100293938686 _0.00057288268896462887
>>> _0.0017724766597981279 _0.00012756882532660762
>>> 0.0011266169145724698 3.9439332284661476e_5 0.00075421374633090137
>>> _0.00031140912439777405 _0.00038153100293938501
>>> 3.9439332284660466e_5 0.00064160958749499744 1.6408293152415618e_5
>>> _0.0028261600905336603 _0.00057288268896460545
>>> 0.00075421374633090712 1.6408293152416783e_5 0.0042952959863754138
>>> ]f=:e mp W
>>> _0.33363012653942103
>>> _0.21892822215417146
>>> _0.10546286817415873
>>> _0.24545184018518121
>>> _0.1044605740131157
>>> ]g=:NbI mp CT
>>> 0.0013909444659834052
>>> 0.00028633175856724745
>>> _0.00036699466085721166
>>> _5.9456044547029092e_6
>>> _0.0013142105194193007
>>> ]h=:g mp NcI
>>> 2.4247689595973174e_7
>>> 4.991488713605179e_8
>>> _6.3976476685240161e_8
>>> _1.0364696420582438e_9
>>> _2.291002230352313e_7
>>> ]i=:g mp Wx
>>> 0.00069547223299170258 0.00014316587928362372 _0.00018349733042860583
>>> _2.9728022273514546e_6 _0.00065710525970965033
>>> ]dX=:f + i NB. step by step solution
>>> _0.33293465430642932
>>> _0.21878505627488784
>>> _0.10564636550458734
>>> _0.24545481298740857
>>> _0.10511767927282535
>>> NB.=======================================================================================
>>>
>>> NB. VERSION 1
>>> NB. dX=(NbI-NbI*CT*NcI*C*NbI)*W+NbI*CT*NcI*Wx
>>> NB. dX=((NbI-(((NbI*CT)*NcI)*C*NbI))*W)+((NbI*CT)*NcI*Wx)
>>> ]dX=:((NbI-(((NbI mp CT) mp NcI) mp C mp NbI)) mp W)+((NbI mp
>>> CT) mp NcI mp Wx)
>>> _0.33363000530097303
>>> _0.2189281971967279
>>> _0.10546290016239707
>>> _0.24545184070341602
>>> _0.10446068856322722
>>> NB.=======================================================================================
>>>
>>> NB. VERSION 2
>>> NB. dX=(((NbI-(((NbI*CT)*NcI)*C)*NbI)*W)+(NbI*CT)*NcI)*Wx
>>> ]dX=:(((NbI-(((NbI mp CT) mp NcI) mp C) mp NbI) mp W)+(NbI mp
>>> CT) mp NcI) mp Wx
>>> _0.16681494203126254 _0.10946408611964216 _0.052731466075317708
>>> _0.12272592061082542 _0.052230401556669366
>>> NB.=======================================================================================
>>>
>>> NB. VERSION 3
>>> NB. dX=((NbI-(((NbI*CT)*NcI)*C)*NbI)*W)+(NbI*CT)*NcI*Wx
>>> ]dX=:((NbI-(((NbI mp CT) mp NcI) mp C) mp NbI) mp W)+(NbI mp
>>> CT) mp NcI mp Wx
>>> _0.33363000530097303
>>> _0.2189281971967279
>>> _0.10546290016239707
>>> _0.24545184070341602
>>> _0.10446068856322722
>>> NB.=======================================================================================
>>>
>>> NB. VERSION 4
>>> NB. dX=(NbI-NbI*CT*NcI*C*NbI)*W+NbI*CT*NcI*Wx
>>> NB. D=(NbI*CT)*NcI
>>> NB. dX=(NbI-D*C*NbI)*W+D*Wx
>>> NB. dX=((NbI-((D*C)*NbI))*W)+D*Wx
>>> ]dX=:(NbI-((D mp C) mp NbI)) mp W + D mp Wx[D=:(NbI mp CT) mp
>>> NcI
>>> _0.33363012560096128
>>> _0.21892822202521539
>>> _0.10546286851468895
>>> _0.24545184023593167
>>> _0.10446057488621374
>>> NB.=======================================================================================
>>>
>>> NB. VERSION 5
>>> NB. dX=(NbI-NbI*CT*NcI*C*NbI)*W+NbI*CT*NcI*Wx
>>> NB. dX=:(NbI-NbI mp CT mp NcI mp C mp NbI) mp W+NbI mp CT mp NcI
>>> mp Wx
>>> ]dX=:(NbI-NbI mp CT mp NcI mp C mp NbI) mp W+NbI mp CT mp NcI
>>> mp Wx
>>> _0.33363012560096128
>>> _0.21892822202521539
>>> _0.10546286851468895
>>> _0.24545184023593167
>>> _0.10446057488621374
>>> NB.=======================================================================================
>>>
>>> NB. ORDER OF EVALUATION J Primer tell
>>> NB. J has a right-to-left order of evaluation.
>>> NB. J always parenthesies from right-to-left.
>>> NB. Remember: no verb precedence and right-to-left evaluation
>>> NB.=======================================================================================
>>>
>>> NB. How can solve the above matrix equation?
>>> NB.
>>> NB. Can somebody tell me the right solution?
>>> NB.
>>> NB. Thanks, Erik
>>> ----------------------------------------------------------------------
>>
>
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