svd

0fe7bc08-836c-40af-848a-8ecfdb19f32f 56 4 5 false false 0 decimal

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

svd

Singular Value Decomposition with initials Uo, So, Vo and tolerances T and t..A = U S V' A T t U.o S.o V.o MSVD U U.o : S S.o transpose : V V.o : m A rows : n A cols : DS 0 : 16 mat UB LB 12 mat m n 12 mat max m n 12 mat min 12 mat eval : r 1 m range : c 1 n range : s 1 m n * (range : p 1 LB range : 14 mat M v z M v el M v el M v el abs t > (* : 11 line : S μ ν qr S S transpose : Id μ identity : Q Id : 13 mat χ μ ν > 1 ν range 1 μ 1 - (range if ρ χ μ range : v μ 1 matrix : 12 mat y S ρ χ el : y 1 el y 1 el S χ χ el sign y norme * + : 12 mat w y y norme / : v ρ el w ρ 1 + χ - el : 12 mat H Id 2 v * v transpose * - : S H S * : Q Q H * : 13 mat 41 line for Q 31 line : Iter 110 T log10 * UB * (- range UT S : UT c r el UT c r el c r < (* : norme T DS p el S p p el : norme * < break U U S m n qr * : V V S n m qr * : S s z 13 mat if 21 line for k p S k k el 0 < S k k el S k k el - : U r k el U r k el - : 12 mat continue if for U 1 m 2^range z S transpose V 1 n 2^range z 13 mat eval 81 line :

Singular Value Decomposition A TOL tol MSVD A TOL tol A rows identity A A cols identity MSVD 11 line :

Singular Value Decomposition A MSVD A 0.1 TOL * TOL MSVD 11 line :

svd Examples

TOL 109 -^:

Ex 1

A 3111 - 3123 mat :

U S V 13 mat A MSVD :

A U S * V transpose * - norme 2.4 109 -^*

U S V 13 mat 0.7071 0.7071 - 0.7071 0.7071 22 mat 3.4641 000 3.1623 023 mat 0.4082 0.8944 - 0.1826 - 0.8165 0.4472 0.3651 - 0.4082 0 0.9129 33 mat 13 mat

Ex 1'

A A transpose :

U S V 13 mat A MSVD :

A U S * V transpose * - norme 2.1 109 -^*

U S V 13 mat 0.4082 0.8944 - 0.1826 - 0.8165 0.4472 0.3651 - 0.4082 0 0.9129 33 mat 3.4641 00 3.1623 0032 mat 0.7071 0.7071 - 0.7071 0.7071 22 mat 13 mat

Ex 2

A 322232 - 23 mat :

U S V 13 mat A MSVD :

A U S * V transpose * - norme 1.1 109 -^*

U S V 13 mat 0.7071 - 0.7071 0.7071 - 0.7071 - 22 mat 50003023 mat 0.7071 - 0.2357 0.6667 - 0.7071 - 0.2357 - 0.6667 0 0.9428 0.3333 33 mat 13 mat

Ex 2'

A A transpose :

U S V 13 mat A MSVD :

A U S * V transpose * - norme 5.8 1010 -^*

U S V 13 mat 0.7071 0.2357 - 0.6667 - 0.7071 0.2357 0.6667 0 0.9428 - 0.3333 33 mat 50030032 mat 0.7071 0.7071 - 0.7071 0.7071 22 mat 13 mat

Ex 3

A 92991815988071416481882022858719213869325291724768390235828991796139597101294967811181007784105 mat :

U S V 13 mat A MSVD :

A U S * V transpose * - norme 1.4 109 -^*

S 363.4811 00000 231.5016 00000 117.8766 00000 22.4612 0000 2.8156 108 -^* 17.7241 0000000000000000000000000105 mat

U 0.2226 0.4705 0.0536 - 0.4124 0.2407 - 0.2459 - 0.3826 0.2205 - 0.2094 0.4459 0.2282 0.4178 0.1317 - 0.1262 - 0.2822 0.0235 - 0.3761 - 0.6485 - 0.2416 - 0.199 - 0.2565 0.0734 0.6669 0.2445 0.2193 - 0.3739 - 0.1709 - 0.1731 0.2849 - 0.3099 - 0.2291 0.3978 0.0278 0.5274 - 0.3461 - 0.3566 0.1289 0.2819 0.3719 - 0.1704 0.2258 0.4274 0.1097 0.0248 - 0.4538 0.1425 0.0068 0.4047 0.5421 0.2642 - 0.3809 0.2089 - 0.02 0.4446 0.2286 - 0.7122 0.0954 - 0.1468 - 0.0721 0.1338 - 0.3865 0.2617 - 0.0581 - 0.094 - 0.2943 0.0857 - 0.723 0.1132 - 0.2371 - 0.2903 - 0.3703 0.0538 - 0.6888 - 0.1533 0.1458 - 0.3055 - 0.2595 - 0.3868 0.1133 - 0.1327 - 0.3874 0.2816 - 0.1014 0.4953 - 0.334 - 0.2093 - 0.1142 - 0.2422 - 0.5366 0.0046 0.3841 0.2521 - 0.1833 0.0073 0.4659 0.0047 - 0.2325 - 0.1223 0.1494 - 0.6685 1010 mat

V 0.3727 0.5861 0.473 - 0.3459 - 0.4174 0.349 0.6295 0.4358 0.3409 0.4192 - 0.4805 0.2661 - 0.6503 0.3274 - 0.4102 0.5043 0.3105 - 0.2819 - 0.3825 - 0.6507 - 0.504 0.3048 - 0.2898 - 0.7146 0.2416 55 mat

Ex 4

Check the time and accuracy and compare with other softwre. As expected, isn't very fast.

A 1520 matrix :

r 1 A length range :

A r el 200109 -^109^random * * 100 - eval :

TOL 105 -^:

tol 104 -^:

t.o 0 time :

U S V 13 mat A TOL tol MSVD :

0 time t.o - 145.79 s *

SUV A dn_LinAlgSVD :

U.dn S.dn V.dn 13 mat SUV 2 el SUV 1 el SUV 3 el transpose 13 mat :

A U S * V transpose * - norme 0

U U.dn - norme 5.7

S S.dn - norme 0

A U.dn S.dn * V.dn transpose * - norme 31012 -^*

V V.dn - norme 6.1

Alvaro