<p> Explore Pseudo-pieces </p>

b81186be-89e3-4ba1-ad36-a1c2d47e0ccf 15 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Chemical balance [5.2]

x₁[4Yb(CN)₃*3Yb(CN)₂]+x₂CsRu(CN)₂F₂+x₃CsRu(CN)₄+x₄CsHF₂+x₅[PtF₃*7H₂O]

=x₆[Pt(NH₃)₂(C₅H₄ON]₂(NO₃)₂*2H₂O+x₇Cs_3.99Yb(CN)₆+x₈HRuF_2.97+x₉NO₂

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3 appVersion 0.98.6179 ≡

This document solves the chemical balance from Risteski paper. It solves for the fractional weight of the components at left Offered pieces wise & in program Line 1 balance [+] line 2 balance [-]

x 1 el x 2 el x 3 el # # # # # # 19 mat

Reaction 
Matrix A 7000001 - 0018240010 - 6 - 001824008 - 6 - 01 - 011100 3.99 - 0001100001 - 00202300 2.97 - 000011424 - 01 - 0000012 - 0000000710 - 002 - 99 mat :

I A rows identity :

a 1 el B 1 el 12 mat A tr - A a 1 el I * + 12 mat :

a 2 el A B 1 el * tr - 2 / : B 2 el A B 1 el * a 2 el I * + : 21 line

a 3 el A B 2 el * tr - 3 / : B 3 el A B 2 el * a 3 el I * + : 21 line

a 4 el A B 3 el * tr - 4 / : B 4 el A B 3 el * a 4 el I * + : 21 line

a 5 el A B 4 el * tr - 5 / : B 5 el A B 4 el * a 5 el I * + : 21 line

a 6 el A B 5 el * tr - 6 / : B 6 el A B 5 el * a 6 el I * + : 21 line

a 7 el A B 6 el * tr - 7 / : B 7 el A B 6 el * a 7 el I * + : 21 line

a 8 el A B 7 el * tr - 8 / : B 8 el A B 7 el * a 8 el I * + : 21 line

Matrix 'A' solved x A° A B 7 el (2^* a 8 el (2^/ : i 1 A rows range u i el 1 : for I A° A * - (u * 31 line :

x 0.0485 0.6235 0.3535 0.3785 0.2993 0.1496 0.3397 0.977 0.2993 91 mat

collect pieces
PseudoInverse i 18 range pseudo i el B i el : for

system pseudo x stack :

<p> Explore Pseudo-pieces </p>

pseudo 11 mat 1 el 5 - 000001 - 001810 - 40010 - 6 - 001828 - 008 - 6 - 01 - 01111 - 00 3.99 - 00011012 - 001 - 00202312 - 0 2.97 - 000011424 - 12 - 1 - 0000012 - 012 - 00000710 - 0014 - 99 mat 11.05 001 - 14 - 2451018 14.05 24 - 26 - 114 - 21218 35.7 4 - 1828 - 22.05 22 - 115 - 19818 29.76 10367 - 3 - 31.06 55.86 - 77.76 31.89 3.99 1 - 368 - 4 - 0 45.05 16 - 12 - 121 - 3615 - 1322 - 38.97 - 31.99 19.98 - 32.64 0033 - 1559 - 241 - 290 42.06 69.28 003 - 14 - 18 - 240 50.99 0013 - 720 - 128 - 1400 22.7 74.05 99 mat 183.78 - 3315 - 52143122 - 7.06 - 62.28 - 0 53.1 - 2.97 179.8 - 416 895.7 411.9 - 145.44 533.64 - 32 18.9 247.1 421.93 - 392 945.76 487.92 - 105.48 491.06 - 42.05 - 72 110.72 64.8 - 42.66 - 676.73 669.34 - 99.93 - 206.98 - 536 10.95 - 2.95 - 44 - 472.13 - 38636 14.47 6216 0.99 - 68.97 - 22 151.11 - 199.11 63.78 36.06 - 13 - 324 - 244372 - 563.06 1528.12 938 - 82.28 662.36 - 15 - 36 - 2230 - 44 122.99 79.98 - 27.96 314.41 - 1 - 108 - 120172 - 260 961.05 711.9 - 115.8 287.8 - 416.23 - 99 mat 766.69 - 13 - 267 199.06 - 527.12 - 84 131.7 - 226.4 15 3554.64 - 330.01 604.38 262.36 - 490.82 1334.58 - 545.76 - 182.2 115.8 3014.64 - 12.26 834.21 478.36 - 772.45 - 224.46 - 534 - 397.88 506.03 1330.56 612.77 - 817.75 1285.5 - 3579.01 - 2173.46 177.31 - 1411.14 54.8 1.8 228.07 571.73 - 764 1914.24 819.84 - 222.96 710.29 - 9.05 - 252.72 129.19 408.95 - 484 1363.19 571.17 - 115.98 504.03 - 94.97 4572 - 40.82 - 1579.18 1230.66 - 2429.44 - 36 958.84 - 1175.46 402396 - 8.97 - 134.99 88 - 169.91 - 12.22 - 91.56 - 282.36 321692 - 306.75 - 1013.05 1048 - 3715.91 - 2134.7 617.4 - 1037.49 1200.23 99 mat 781.17 - 50.18 - 289.82 162.76 - 1260.4 - 552 36.94 409.34 297 - 8063.47 - 558.25 - 3966.02 3477.28 - 10651.58 - 3440.7 1004.88 - 4018.64 835.98 - 5866.03 - 6.79 - 2148.73 1461.28 - 4209.29 - 163.66 155.52 - 1973.1 1846.27 - 13003.56 107.63 - 4044.59 - 2897.78 5832.83 470.78 2568.7 2698.86 - 927.35 - 6173.28 - 351.24 1303.6 652.72 - 98.05 - 1546.82 - 988.2 - 297.73 589.83 3266.64 - 145.33 728.15 542.36 - 70.97 731.8 - 505.32 - 217.21 219.01 4313.52 - 511.38 - 2493.34 2117.5 - 9326.3 - 4416 265.84 3281.36 2383.18 - 503.64 - 30.31 - 246.17 204 - 812.14 - 322.5 9 - 311.43 198.99 - 869.4 918.09 1938.71 - 2604 7199.6 4296.58 - 1635.72 2006.75 - 3399.84 - 99 mat 2049.14 - 160.12 464.6 - 978.18 503.5 552 - 7.26 - 415.98 - 304.18 4895.44 1326.42 - 497.8 - 2399.24 5579.44 - 1705.96 - 1491.24 1437.58 2294.06 - 2507.24 - 1059.36 - 2002.72 1289.48 - 12637.09 - 1127.02 1155.12 - 3878.74 1543.8 16285 1367.73 7878.26 - 5513.56 28183.9 13544.7 - 347.6 9799.75 - 5899.29 13425.86 - 534.74 - 5868.58 4734.56 - 14943.21 - 3281.86 1151.4 - 5680.31 2483.26 - 7143.85 - 188.03 - 3022.53 2511.28 - 7519.6 - 1330.19 189.78 2607.79 1166.17 - 5480.64 1352.12 3515.96 - 6980.34 3568.86 3944 - 23.9 - 4055.06 - 2273.02 360 60.58 152.7 - 432240 - 83.22 - 22.44 1031.17 - 151.81 12285.36 - 830.8 - 5721.12 4353.44 - 15795.3 - 5083.42 5135.64 - 3925.52 7843.91 99 mat 336.11 231.28 - 263.76 133.08 - 44.36 - 80 26.92 - 1143.2 143.76 - 1432.09 9920.16 10477.08 - 478.56 1138.65 1522.26 3523.08 - 9154.9 5085.92 570.25 - 9785.81 - 10007.58 190.56 - 1894.75 900.78 - 1992.12 10444.96 5090.33 - 3194.55 - 6413 - 7655.33 1067.52 - 4272.38 - 1612.92 779.08 11696.26 3920.31 - 2028.2 2446.36 - 1657.61 677.76 2184.19 495.72 - 313.68 6347.49 902.06 - 1014.1 1701.5 - 1307.13 338.88 1092.09 247.86 - 156.84 3379.16 690.19 - 584.64 - 1666.48 - 1893.84 931.56 - 310.52 - 560 188.44 - 8168.72 1030.08 - 861.84 158.68 - 176.48 - 28896 621.48 1530.96 - 20625.44 150.92 - 2028.2 201.27 - 587.47 - 677.76 2184.19 495.72 - 313.68 5833.19 1248.22 - 99 mat 0 47.52 47.52 - 0000 166.32 - 23.76 0 610.48 610.48 - 0000 2136.69 - 305.24 0 346.16 346.16 - 0000 1211.55 - 173.08 0 370.59 370.59 - 0000 1297.08 - 185.3 0 293.02 293.02 - 0000 1025.58 - 146.51 0 146.51 146.51 - 0000 512.79 - 73.26 0 332.64 332.64 - 0000 1164.24 - 166.32 0 956.64 956.64 - 0000 3348.24 - 478.32 0 293.02 293.02 - 0000 1025.58 - 146.51 99 mat 81 mat

j 7 :

Each piece is invertible

A° A B j el (2^* a j 1 + el (2^/ : 0.114 0.032 0.021 - 0.045 - 0.015 - 0.027 0.009 - 0.001 0.007 0.488 4.804 4.993 - 0.163 0.388 0.518 1.199 - 1.876 - 2.445 0.194 - 2.523 - 2.598 0.065 - 0.645 0.307 - 0.678 0.725 1.329 - 1.088 - 1.317 - 1.74 0.363 - 1.454 - 0.549 0.265 0.951 0.902 - 0.69 0.148 - 0.12 - 0.231 0.744 0.169 - 0.107 0.235 - 0.035 0.345 0.237 - 0.103 0.115 0.372 0.084 - 0.053 0.048 - 0.064 - 0.199 - 0.21 0.133 - 0.317 - 0.106 - 0.191 0.064 - 0.061 0.038 0.293 2.181 2.295 - 0.098 0.033 0.212 0.521 - 0.802 - 1.066 0.69 0.616 0.885 - 0.231 0.744 0.169 - 0.107 0.411 - 0.083 - 99 mat

A° invert 1.626 - 5.788 1013^* - 5.788 1013^* 0.307 5.024 2.548 - 2.291 2.026 1014^* 2.894 1013^* - 92.821 - 7.436 1014^* - 7.436 1014^* 3.875 64.578 42.734 - 36.282 2.603 1015^* 3.718 1014^* - 44.838 - 4.216 1014^* - 4.216 1014^* 2.34 36.641 26.561 - 17.975 1.476 1015^* 2.108 1014^* - 67.274 - 4.514 1014^* - 4.514 1014^* 3.422 39.202 19.871 - 21.677 1.58 1015^* 2.257 1014^* - 53.193 - 3.569 1014^* - 3.569 1014^* 1.808 30.998 15.712 - 20.295 1.249 1015^* 1.785 1014^* - 26.596 - 1.785 1014^* - 1.785 1014^* 2.943 18.485 7.856 - 10.147 6.246 1014^* 8.923 1013^* - 60.384 - 4.052 1014^* - 4.052 1014^* 3.222 49.18 41.836 - 23.038 1.418 1015^* 2.026 1014^* - 173.66 - 1.165 1015^* - 1.165 1015^* 6.348 102.073 53.295 - 66.256 4.078 1015^* 5.826 1014^* - 53.193 - 3.569 1014^* - 3.569 1014^* 1.998 38.011 25.712 - 20.295 1.249 1015^* 1.785 1014^* - 99 mat

PseudoInverse  Mass balance
Courtesy: Viktor Korobov {Mathcad 11] x r A rank : I A rows identity : a 1 el A tr - : B 1 el A a 1 el I * + : n 2 r range B n el A B n 1 - el * A B n 1 - el * tr - n / I * + : a n el A B n 1 - el * tr - n / : n n 1 + : 31 line for Ω A B r 1 - el (2^* a r el (2^/ : i 1 A rows range u i el 1 : for I Ω A * - (u * 81 line :

ChemicalBalance Yb C N Cs Ru F H Pt O 91 mat x augment :

ChemicalBalance Yb 0.0485 C 0.6235 N 0.3535 Cs 0.3785 Ru 0.2993 F 0.1496 H 0.3397 Pt 0.977 O 0.2993 92 mat

Is that reaction matrix solvable ? ... Yes

The Reaction Matrix A 7000001 - 0018240010 - 6 - 001824008 - 6 - 01 - 011100 3.99 - 0001100001 - 00202300 2.97 - 0000012 - 00000011424 - 01 - 00000710 - 002 - 99 mat :

PseudoInverse  Mass balance
Courtesy: Viktor Korobov {Mathcad 11] x r A rank : I A rows identity : a 1 el A tr - : B 1 el A a 1 el I * + : n 2 r range B n el A B n 1 - el * A B n 1 - el * tr - n / I * + : a n el A B n 1 - el * tr - n / : n n 1 + : 31 line for Ω A B r 1 - el (2^* a r el (2^/ : i 1 A rows range u i el 1 : for I Ω A * - (u * 81 line :

Coeffs 209279 x * :

Fractional
mass balance x 0.189221 2.430894 1.378375 1.475676 1.166796 0.583398 1.324548 3.809269 1.166796 91 mat

... Integer 
mass balance Coeffs 3960050873528846530882824418612209327720079720024418691 mat

Components Yb C N Cs Ru F H Pt O 91 mat

Example 1 ....................

A 00101424 - 01 - 011000001 - 002203003 - 0111000 3.99 - 000007001 - 0042018010 - 6 - 004201808 - 6 - 01 - 000012 - 0000000710 - 002 - 99 mat :

H Au Cl K Fe C N Pt O 91 mat

missing x9

PseudoInverse  Mass balance
Courtesy: Viktor Korobov {Mathcad 11] x r A rank : I A rows identity : a 1 el A tr - : B 1 el A a 1 el I * + : n 2 r range B n el A B n 1 - el * A B n 1 - el * tr - n / I * + : a n el A B n 1 - el * tr - n / : n n 1 + : 31 line for Ω A B r 1 - el (2^* a r el (2^/ : i 1 A rows range u i el 1 : for I Ω A * - (u * 81 line :

Coeffs 707943 / * x * :

Fractional
mass balance x 0.296 0.549 0.339 0.042 0.253 0.126 0.297 0.845 0.253 91 mat

Integer  mass balance Coeffs 2793.000000 5179.000000 3200.000000 400.000000 2386.000000 1193.000000 2800.000000 7972.000000 2386.000000 91 mat

Components H Au Cl K Fe C N Pt O 91 mat

HEADER Symbol x/frac x.int'r 13 mat :

Table Yb C N Cs Ru F H Pt O 91 mat 0.0485 0.6235 0.3535 0.3785 0.2993 0.1496 0.3397 0.977 0.2993 91 mat 2793.000000 5179.000000 3200.000000 400.000000 2386.000000 1193.000000 2800.000000 7972.000000 2386.000000 91 mat 13 mat :

Table HEADER j i x lastError Table HEADER Coeffs j system pseudo i x B a lastError j i

Table
mat("Yb","C","N","Cs","Ru","F","H","Pt","O",9,1)	mat(97/2000,1247/2000,707/2000,757/2000,2993/10000,187/1250,3397/10000,977/1000,2993/10000,9,1)	mat(2793,5179,3200,400,2386,1193,2800,7972,2386,9,1)
#
Symbol	x/frac	x.int'r
#
#
#

Solving square invertible matrix

Smath rref solving conventional invertible matrix M rrefSmath U M 1 M rows range 1 M rows range el : V M M cols col : sol U 1 -^V * : U rows identity sol augment sol 21 mat 41 line :

M 154548239457518725442626925719856 mat :

M rrefSmath 1 el 10000 1.9787 01000 1.3394 00100 0.0266 00010 0.1054 00001 0.3273 - 56 mat

N M 14 range 15 range el :

N rrefSmath 2 el 1.7897 2.3452 - 0.4643 - 3.1587 41 mat

R 78010 Random.N : 4874831741000256621041096903083107314998591733310013510895072878 mat

R rrefSmath 1 el 1000000 5.2612 0100000 0.3368 - 0010000 6.5755 - 0001000 2.2165 0000100 2.9816 0000010 2.2611 0000001 4.8243 - 78 mat

R rrefSmath 2 el 5.2612 0.3368 - 6.5755 - 2.2165 2.9816 2.2611 4.8243 - 71 mat

Solve matrix observer

select rectangular matrix M R 36 range 28 range el : 1041096900831073199859173310013547 mat

select a square matrix to be solved U M 1 M rows range 1 M rows range el :

rows(M) < j ... solve 'U' wrt col 'j' => right of square 'U' j 6 :

V M j col (: 931341 mat

sol U 1 -^V * (: 0.0108 - 0.7782 - 0.5367 0.7616 41 mat

Matrix ...solutions ... col solved for U sol U sol * 13 mat 1041090831099853310044 mat 0.010838 - 0.778227 - 0.53672 0.761566 41 mat 931341 mat 13 mat

Traditional invertible matrix solve compactum

U 107 8.1 7.2 7.08 5.04 658 5.98 9.89 9 6.99 4.99 9 9.98 44 mat : V 3223333141 mat : wxyz U 1 -^V * : 15 mat

[w,x,y,z] 
solutions  
... sanity wxyz U wxyz * 12 mat 81 - 13734 - 2241 mat 3223333141 mat 12 mat