Matrix svds(data)

8fe434d4-afcf-4dca-a3c0-7f2156039788 70 4 5 decimal

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

Matrix svds(data)

data svds Unwrap matrix in vector N N data 1 col : i 2 data cols range N N data i col stack : for N 31 line Strip 'N' of 0's k 0 : i 1 N rows range N i el 0 ≠ R k k 1 + : el N i el : 11 line end of strip if 11 line for R diag 61 line :

DelColRow(M,c) ... alternate as it may apply

M col DelCol N 0 : ct 1 : n 1 M cols range n col ≠ N 1 ct el M n col : ct ct 1 + : 21 line Unest 'N' if for v N 1 el : j 2 N cols range v v N j el augment : for v 61 line :

M row DelRow N 0 : ct 1 : n 1 M rows range n row ≠ N 1 ct el M n row : ct ct 1 + : 21 line Unest 'N' if for v N 1 el : j 2 N rows range v v N j el stack : for v 61 line :

Part_1 Greville Pseudo-Inverse from dn_LinAlgSVD(A)

THE TWO "SVD" COMPANIONS: The M1 is regular for any rectangular or square matrix. The companion svds(data) returns the svds vector. Associated with diag(vector}, we get the square svds ... matrix that produces the pseudo-inverse.

3 appVersion 0.98.6179 ≡

A 121 - 2.3 442 - 5.1 10 0.8 643 mat :

M A dn_LinAlgSVD :

M 8.6574 000 5.1743 000 3.0357 00043 mat 0.0707 - 0.3774 0.4434 - 0.8099 0.671 - 0.0211 0.6192 - 0.4074 - 0.4484 - 0.7004 0.5516 0.0635 - 0.5862 - 0.6054 - 0.3402 0.4172 44 mat 0.0828 - 0.6447 - 0.7599 - 0.1884 - 0.7589 0.6233 - 0.9786 - 0.0915 - 0.1843 33 mat 31 mat

U M 2 el (14 range 13 range el : 0.07072235 - 0.37740198 0.44339051 - 0.6709905 - 0.0211077 0.61916248 - 0.44842242 - 0.70041032 0.55163942 0.58624861 - 0.60543174 - 0.34020085 43 mat

V M 3 el : 0.08283698 - 0.6446899 - 0.75994274 - 0.18840711 - 0.75894287 0.62330449 - 0.97859123 - 0.09154596 - 0.1843327 33 mat

1/1 sanity Mathcad svds M 1 el svds : 8.657433 000 5.174279 000 3.035734 33 mat

<= sanity check => U svds * V * 121 - 2.3 442 - 5.1 1045 / 643 mat

A 121 - 2.3 442 - 5.1 10 0.8 643 mat

svds V * 0.717 - 5.581 - 6.579 - 0.975 - 3.927 3.225 - 2.971 - 0.278 - 0.56 33 mat

Greville Pseudo-Inverse ?

MCDsvd U V transpose stack :

MCDsvd 0.070722349587322 - 0.37740198210046 0.443390513435775 - 0.67099049883796 - 0.021107703127728 0.619162481705024 - 0.448422418013542 - 0.700410324359898 0.551639424495536 0.58624861173475 - 0.605431735462701 - 0.340200851242717 0.082836981864425 - 0.188407113592162 - 0.97859122925942 - 0.644689898355372 - 0.758942865418423 0.091545955612544 - 0.759942740865476 - 0.623304489457876 - 0.18433269928996 73 mat

U U sqrt vectorize :

m M 1 el :

New svds m U * vectorize svds 2.30232963718658 i * - 000 0.751744607301144 000 2.25471379550402 33 mat

Implicit solve

x y z f 5 x * 6 y z + (* - 8 - 3 x y - (* 2 z * + 6 + x y + z + 2 - 31 mat :

sol x y z f x y z 31 mat roots : 1.818182 2.363636 2.181818 - 31 mat

x y z 31 mat sol 1 el sol 2 el sol 3 el 31 mat :

sanity x y z f 0.000000000000031 0.00000000000003 - 0.00000000000002 - 31 mat

Implicit solve

x y z f x 2 y * - z + 14 - x y - z - 2 + 2 x * 3 y * + 4 z * + 28 - 31 mat :

sol x y z f x y z 31 mat roots : 32 - 731 mat

x y z 31 mat sol 1 el sol 2 el sol 3 el 31 mat :

sanity x y z f 0.000000000000000 0.000000000000000 0.000000000000000 31 mat

data 35.7988 000000 13.8098 000000 4.1351 000000 3.6737 000000 3.023 056 mat :

data svds 35.7988 00000 13.8098 00000 4.1351 00000 3.6737 00000 3.023 55 mat

A 0.3 0.2 6.6 1.1 - 10 4.5 1.8 - 0.3 - 6.5 5 7.3 - 9.7 10.9 4.1 - 2 8.1 2.7 - 8.7 8.9 6 - 45 mat :

Example for Sanity check Mathcad 11 sol A dn_LinAlgSVD : 19.28 00000 16.064 00000 11.9252 00000 3.8146 045 mat 0.3055 0.3679 0.7084 - 0.5192 0.2626 - 0.2338 0.548 - 0.759 - 0.7828 0.4179 0.2946 0.3548 - 0.4743 - 0.7971 0.3333 0.169 44 mat 0.5522 - 0.4879 0.3372 0.4914 - 0.3192 0.2844 0.0967 0.862 0.4044 0.0561 0.1785 - 0.235 0.1342 0.0859 - 0.9421 - 0.1833 0.6364 - 0.3296 0.6674 - 0.0857 - 0.7408 - 0.5407 - 0.1288 0.3771 0.0105 - 55 mat 31 mat

Diag sol 1 el A cols DelCol (: 19.28 0000 16.064 0000 11.9252 0000 3.8146 44 mat

Vmcd sol 2 el :

Umcd sol 3 el transpose 5 DelCol : 0.5522 - 0.2844 0.1785 - 0.1833 0.4879 0.0967 0.235 0.6364 - 0.3372 0.862 0.1342 0.3296 0.4914 - 0.4044 0.0859 - 0.6674 - 0.3192 0.0561 0.9421 - 0.0857 - 54 mat

1/1 sanity Smath MCD below.
many sign [±] disagree ? MCD11 A transpose svd ≡

MCD Diag Umcd Vmcd stack :

MCD 19.28 0000 16.064 0000 11.9252 0000 3.8146 0.5522 - 0.2844 0.1785 - 0.1833 0.4879 0.0967 0.235 0.6364 - 0.3372 0.862 0.1342 0.3296 0.4914 - 0.4044 0.0859 - 0.6674 - 0.3192 0.0561 0.9421 - 0.0857 - 0.3055 0.3679 0.7084 - 0.5192 0.2626 - 0.2338 0.548 - 0.759 - 0.7828 0.4179 0.2946 0.3548 - 0.4743 - 0.7971 0.3333 0.169 134 mat

Part_2 Solving singular matrix

Pseudo-Inverse . . . Scientific/Engineering

Example 1 . . . Piecewise/Program

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The Reaction Matrix ... NOT invertible A 7000001 - 0018240010 - 6 - 001824008 - 6 - 01 - 011100 3.99 - 0001100001 - 00202300 2.97 - 0000012 - 00000011424 - 01 - 00000710 - 002 - 99 mat :

I A rows identity :

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

B 1 el 4 - 000001 - 00189 - 40010 - 6 - 001827 - 008 - 6 - 01 - 01110 - 00 3.99 - 00011011 - 001 - 00202311 - 0 2.97 - 0000012 - 11 - 0000011424 - 012 - 00000710 - 0013 - 99 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

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 0.1892 2.4309 1.3784 1.4757 1.1668 0.5834 1.3245 3.8093 1.1668 91 mat

is the 'x' solving pseudoinverse
associated with (I-A°*A)*u A° A B 7 el (2^* a 8 el (2^/ ≡

A° A B 7 el (2^* a 8 el (2^/ : 0.2231 0.5335 0.5647 - 0.0463 0.0154 0.0319 0.421 - 0.1101 - 0.2666 1.8841 11.9067 12.6394 - 1.3404 0.7801 0.5776 9.6145 - 2.4956 - 6.1171 0.5978 1.1485 1.381 - 0.6028 0.8676 0.273 - 2.4155 - 0.0568 - 0.5756 0.2398 - 2.0864 1.9932 - 0.3514 1.2162 - 0.5851 0.5679 - 0.5216 - 0.8736 1.3608 3.2757 3.8049 - 0.7959 0.932 0.1403 - 4.0007 - 0.5154 - 1.8052 0.6804 1.4171 1.6817 - 0.398 0.466 0.0701 - 1.7277 - 0.2577 - 0.7922 0.5619 3.6716 3.8901 - 0.3244 0.1081 0.223 2.7259 - 0.7704 - 1.8347 2.4819 12.6663 13.6315 - 1.9432 0.6477 0.3046 10.6688 - 2.5524 - 6.4983 1.3608 4.185 4.7143 - 0.7959 0.932 0.1403 - 4.6833 - 0.5154 - 1.7599 99 mat

PseudoInverse  Mass balance
Courtesy: Viktor Korobov [Mathcad 11]
from Ice B. Risteski [U.T.] paper 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 :

209279 ... iterated manual.
205002 ... already very close. 205002

Coeffs 209279 x * :

209279

205002

.......... Components ..........

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 39600.00000 508735.00000 288465.00000 308828.00000 244186.00000 122093.00000 277200.00000 797200.00000 244186.00000 91 mat

Yb C N Cs Ru F H Pt O 91 mat

The integer 209279 minimizes Σq q Coeffs 0 round Coeffs - vectorize :

q 81011 -^* 3106 -^* 2106 -^* 6107 -^* - 3107 -^* 2107 -^* 1108 -^* - 8107 -^* 5107 -^* 91 mat

q sum 6.107424 106 -^*

U x transpose 11 mat :

U 0.18922 2.43089 1.37838 1.47568 1.1668 0.5834 1.32455 3.80927 1.1668 19 mat :

U 946150000 / 243089100000 / 6891950000 / 92236250 / 29172500 / 29175000 / 2649120000 / 380927100000 / 29172500 / 19 mat

A dn_LinAlgSVD 35.708 000000000 25.008 000000000 5.149 000000000 4.243 000000000 2.45 000000000 1.901 000000000 0.398 000000000 0.143 000000000 1.205 1016 -^* 99 mat 0.109 - 0.215 0.016 0.484 - 0.345 0.547 0.524 0.119 5.496 1017 -^* - 0.525 - 0.451 0.057 0.098 0.304 - 0.015 0.149 - 0.569 0.263 - 0.484 - 0.484 0.046 - 0.004 - 0.181 0.217 - 0.055 - 0.615 - 0.263 0.025 - 0.046 0.453 - 0.765 0.278 0.303 0.188 0.043 1.174 1016 -^* 0.005 - 0.008 0.216 - 0.007 0.31 - 0.532 - 0.756 0.055 3.913 1016 -^* 0.032 - 0.055 - 0.86 - 0.406 - 0.099 - 0.005 0.282 - 0.044 3.42 1016 -^* - 0.051 - 0.051 - 0.031 0.02 0.037 - 4.048 105 -^* 0.034 - 0.382 0.919 0.631 - 0.651 - 0.042 0.022 0.254 - 0.253 0.101 0.191 - 5.699 1016 -^* 0.274 - 0.292 - 0.008 0.066 - 0.711 0.464 - 0.053 - 0.307 0.131 - 99 mat 0.53 - 0.059 - 0.114 - 0.02 - 0.305 - 0.76 0.175 0.02 0.029 0.733 0.072 0.152 0.029 - 0.455 - 0.411 0.24 - 0.032 0.004 0.06 0.46 - 0.122 - 0.414 - 0.369 - 0.264 - 0.335 0.53 0.006 0.399 - 0.035 0.271 0.006 - 0.319 - 0.202 - 0.739 - 0.277 0.032 0.079 0.195 - 0.215 - 0.071 - 0.444 0.267 0.291 - 0.351 0.654 - 0.107 0.327 - 0.544 - 0.297 0.161 0.08 0.288 - 0.139 0.602 0.011 0.071 - 0.325 0.693 - 0.425 0.22 0.129 - 0.047 - 0.402 0.011 - 0.654 0.603 - 0.424 - 0.108 - 0.082 - 0.095 - 0.043 0.006 0.035 0.449 0.254 0.272 0.215 0.108 0.244 0.703 0.215 99 mat 31 mat

A dn_LinAlgSVD kind of Pseudo_Inverse ≡

Pseudo-Inverse . . . Scientific/Engineering

Example 2 . . . Program

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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

A tr 15 -

Symbolic A rank 8

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 * :

707943 / * 9439

Fractional mass balance

Integer mass balance

Components

x 0.2959 0.5487 0.339 0.0424 0.2528 0.1264 0.2967 0.8446 0.2528 91 mat

H Au Cl K Fe C N Pt O 91 mat

Coeffs 2793.000000 5179.000000 3200.000000 400.000000 2386.000000 1193.000000 2800.000000 7972.000000 2386.000000 91 mat

x transpose 11 mat 0.2959 0.5487 0.339 0.0424 0.2528 0.1264 0.2967 0.8446 0.2528 19 mat 11 mat

U 0.2959 0.5487 0.339 0.0424 0.2528 0.1264 0.2967 0.8446 0.2528 19 mat :

Symbolic U 295910000 / 548710000 / 3391000 / 531250 / 158625 / 79625 / 296710000 / 42235000 / 158625 / 19 mat

GCD ... LCM

Adapted from Andrey Ivashov a b gcd x a abs : y b abs : 12 mat x 0 ≠ (y 0 ≠ (& x y > x x y mod : y y x mod : if while x y + 31 line :

c 2967 :

d 158 :

a 21 :

b 6 :

a b lcm a b * abs a b gcd / eval :

c d lcm c d * abs c d gcd / eval :

a b gcd a b lcm 21 mat 34221 mat

c d gcd c d lcm 21 mat 1 4.6879 105^* 21 mat

a b c d Clear 1

A 17211411139033 mat :

S A dn_LinAlgSVD : 21.3619 000 8.814 000 5.2899 33 mat 0.234 - 0.2271 0.9453 - 0.6901 - 0.7237 - 0.003 - 0.6849 - 0.6517 0.3261 33 mat 0.7831 - 0.4944 - 0.3773 - 0.0837 0.5173 0.8517 - 0.6163 0.6985 - 0.3637 - 33 mat 31 mat

sanity Mathcad svd(A) S 2 el S 3 el transpose stack 0.234 - 0.2271 0.9453 - 0.6901 - 0.7237 - 0.003 - 0.6849 - 0.6517 0.3261 0.7831 - 0.0837 0.6163 0.4944 - 0.5173 0.6985 - 0.3773 - 0.8517 - 0.3637 - 63 mat

A dn_LinAlgEigenvalues 18.5597 6.7799 - 2.7745 i * + 6.7799 - 2.7745 i * - 31 mat

A dn_LinAlgEigenvectors 0.355 0.459 - 0.28 i * - 0.459 - 0.28 i * + 0.719 0.681 0.681 0.597 0.208 - 0.452 i * + 0.208 - 0.452 i * - 33 mat

eigenall

Function for calculating the coefficients of the characteristic polynomial, eigenvalues and eigenvectors Adapted from: "Introduction to the use of SMath Studio" G. Urroz, May 2010

A eig n A rows : p n 1 matrix : I n identity : p n 1 + 1 el 1.0 - : B A : p n 1 el B tr : j n 1 - 1 n 2 - range B A B p j 1 + 1 el I * - (* : p j 1 el B tr n j - 1 + / : 21 line for p 1 - (n^(- (p * : λ p polyroots : V n 1 - 1 matrix : j 1 n range A.L A λ j el I * - : A.Lm A.L 1 n 1 - 1 n 1 - submatrix : b.L A.L 1 n 1 - n n submatrix : x.L A.Lm 1 -^b.L * : V x.L V augment : 51 line for V V 1 n 1 - 1 n submatrix : V.b 1 n matrix : j 1 n range V.b 1 j el 1 : for V V V.b stack : j 1 n range x.L V j col : x.L x.L x.L norme / : V V x.L augment : 31 line for V V 1 n n 1 + 2 n * submatrix : p λ V 31 sys 181 line :

12 - 422 - 22 - 642 - 38268444 mat eig 1.3 103^* - 530 - 93101 - 51 mat 1.95 - 6.33 7.83 - 13.45 41 mat 0.35 - 0.1 0.79 0.7 - 0.21 - 0.51 0.52 - 0.46 0.59 - 0.57 0.26 0.54 - 0.69 0.64 0.18 0.11 44 mat 31 sys

2 - i + 0002 - i - 0001 - 33 mat eig 5 - 9 - 5 - 1 - 41 mat 1 - 2 - i + 2 - i - 31 mat 00000011133 mat 31 sys

1 - 004 - 22 mat eig 2 el 1 - 4 - 21 mat

ε 106 -^:

EIGENVECTORS CALCULATION FAILS FOR 2x2 MATRICES:

1 - ε + 0 ε + 0 ε - 4 - 22 mat eig 3 el 3.33 107 -^* 11 3.35 107 -^* 22 mat

1 - 004 - 22 mat eig 1 el 45131 mat

1 - i - 001 - i + 22 mat eig 1 el 22131 mat

1 - i - 001 - i + 22 mat eig 2 el 1 - i + 1 - i - 21 mat

A eig 99619851 - 41 mat 6.78 - 2.775 i * + 6.78 - 2.775 i * - 18.56 31 mat 0.355 - 0.062 0.534 i * + 0.062 0.534 i * - 0.719 - 0.285 0.618 i * - 0.285 0.618 i * + 0.597 0.497 0.497 33 mat 31 sys

A dn_LinAlgSVD 21.362 000 8.814 000 5.29 33 mat 0.234 - 0.227 0.945 - 0.69 - 0.724 - 0.003 - 0.685 - 0.652 0.326 33 mat 0.783 - 0.494 - 0.377 - 0.084 0.517 0.852 - 0.616 0.699 - 0.364 - 33 mat 31 mat