dd63489a-78ad-4cc5-86f8-09b40e0e6e45 54 2 5 false false 1 decimal

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

Echelon form of a given matrix_M echel A_M : m A rows : n A cols : r 1 m range : c 1 n range : nxp 1 : 16 mat gen 1 n range p nxp : k 0 : cr p k + : 13 mat A p gen el 0 ≡ k k 1 + : cr p k + : 12 mat cr m > break nothing if temp A p c el : A p c el A cr c el : A cr c el temp 1 c el : 13 mat 31 line while pr A p c el : pr gen el 0 ≠ prm pr pr gen el / : nxp p 1 + m 12 mat min : 12 mat 11 line prm pr : np p : 12 mat 11 line if p 1 > r1 1 p 1 - (range : A r1 c el A r1 c el : 12 mat 11 line nothing 11 line if A p c el prm 1 c el : p m < r3 p 1 + (m range : temp A r3 c el A r3 gen el prm 1 c el * - : 12 mat j 1 r3 length range A p j + c el temp j c el : for 21 line nothing 11 line if prm gen el 1 ≡ (p m ≡ (& break continue if 81 line for A 31 line :

Row Reduced Echelon Form (RREF)_M rref A_M echel : m A rows : n A cols : r 1 m range : c 1 n range : s 1 : 16 mat A s c el 1 n Zeros ≠ (s m ≤ (& s s 1 + : while rp 1 s 1 - (range : pivot A rp c el : mp rp length : 13 mat j rp k c pivot j k el 0 ≠ pos j el k : break 21 line continue 11 line if 11 line for 11 line for mp 1 > j mp 1 - (1 range temp pivot j c el pivot j pos j 1 + (mp range el el pivot j 1 + (mp range c el * - : pivot j c el temp 1 c el : 21 line for 11 line nothing 11 line if rrefA m mp ≠ pivot m mp - n Zeros stack pivot if : rrefA 71 line :

Approximately equal to zero n aez tol 1010 -^: n abs tol < 0 n if 21 line :

randi function as in Octave_rang_m_n randi i 1_m range j 1_n range a i j el_rang 1 el_rang 2 el 100 *_rang 1 el 100 * - random 100 / 0 round + : for for a 21 line :

ROUND -5

ROUND -4

ROUND -3

ROUND -2

ROUND -1

ROUND 0

A 12130846233 mat :

A 1012308 13.2 - 24 mat :

A 1030573242 mat :

A 00130054032543 mat :

A 011022 mat :

A 0 0.6 12 mat 58 randi : 001010100000000010010000000001000000000058 mat

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

rref_Maple A rref maple : 10001000133 mat

rref_Maple A rref maple : 100 5.84 001 3.84 - 24 mat

rref_Maple A rref maple : 1001000042 mat

rref_Maple A rref maple : 10001000100043 mat

rref_Maple A rref maple : 100122 mat

rref_Maple A rref maple : 100100000010101000000100000000000000000058 mat

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

rref_AGR A rref : 10001000133 mat

rref_AGR A rref : 100 5.84 001 3.84 - 24 mat

rref_AGR A rref : 1001000042 mat

rref_AGR A rref : 10001000100043 mat

rref_AGR A rref : 100122 mat

rref_AGR A rref : 100100000010101000000100000000000000000058 mat

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

Δt_Maple toc_Maple tic_Maple - : ms 85

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_Maple toc_Maple tic_Maple - : ms 116

Δt_Maple toc_Maple tic_Maple - : ms 116

Δt_Maple toc_Maple tic_Maple - : ms 85

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_AGR toc_AGR tic_AGR - : ms 16

Δt_AGR toc_AGR tic_AGR - : ms 16

Δt_AGR toc_AGR tic_AGR - : ms 15

Δt_AGR toc_AGR tic_AGR - : ms 16

Δt_AGR toc_AGR tic_AGR - : ms 0

Δt_AGR toc_AGR tic_AGR - : ms 23

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

ROUND 1

ROUND 2

ROUND 3

ROUND 4

ROUND 5

A 1211395308243 - 0462100222002 - 00553318956 mat :

A 11023 - 572 - 512 - 835710 - 20211 - 04 - 45 - 955 mat :

A 6304812051717214 - 02 - 52425715 - 3 - 12 - 10 - 1 - 955 mat :

A 0304812001717214 - 02 - 22405215 - 012 - 10 - 1 - 955 mat :

A 6101 - 3072 - 7 - 48353 - 34 mat :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

rref_Maple A rref maple : 10001 - 001000000103 - 1.5 - 00011 0.5 00000056 mat

rref_Maple A rref maple : 10100011 - 01 - 00011 - 000000000055 mat

rref_Maple A rref maple : 100 0.6279 - 4.3023 - 010 0.0698 - 0.2558 - 001 0.3721 0.6977 000000000055 mat

rref_Maple A rref maple : 010000010000010000010000055 mat

rref_Maple A rref maple : 10 0.1667 0011 - 1 - 000034 mat

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

rref_AGR A rref : 10001 - 001000000103 - 1.5 - 00011 0.5 00000056 mat

rref_AGR A rref : 10100011 - 01 - 00011 - 000000000055 mat

rref_AGR A rref : 100 0.6279 - 4.3023 - 010 0.0698 - 0.2558 - 001 0.3721 0.6977 000000000055 mat

rref_AGR A rref : 010000010000010000010000055 mat

rref_AGR A rref : 10 0.1667 0011 - 1 - 000034 mat

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_Maple toc_Maple tic_Maple - : ms 85

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_Maple toc_Maple tic_Maple - : ms 78

Δt_Maple toc_Maple tic_Maple - : ms 85

Δt_AGR toc_AGR tic_AGR - : ms 16

Δt_AGR toc_AGR tic_AGR - : ms 15

Δt_AGR toc_AGR tic_AGR - : ms 16

Δt_AGR toc_AGR tic_AGR - : ms 5

Δt_AGR toc_AGR tic_AGR - : ms 15

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

ROUND 9

ROUND 6

ROUND 7

ROUND 8

A 0100110110000110111001010110000111001010001001011101010000110100110101514 mat :

A 6101 - 3072 - 7 - 48353 - 34 mat :

A 6304817302 - 51 - 7 - 3 - 43 mat :

A 0000000000001000000001000000000001075 mat :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

tic_Maple 0 time :

rref_Maple A rref maple : 10000 0.5 - 0.5 11 - 0.5 0.5 - 0.5 0.5 0.5 - 01000 1.5 0.5 11 0.5 0.5 - 0.5 0.5 - 1.5 00100 0.5 - 0.5 - 00 0.5 0.5 - 0.5 0.5 0.5 00010 1.5 0.5 01 0.5 0.5 0.5 0.5 - 0.5 00001 0.5 - 0.5 - 00 0.5 - 0.5 0.5 - 0.5 0.5 - 514 mat

rref_Maple A rref maple : 10 0.1667 0011 - 1 - 000034 mat

rref_Maple A rref maple : 10 20.5 01 2.5 - 00000043 mat

rref_Maple A rref maple : 0100000100000100000000000000000000075 mat

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

toc_Maple 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

tic_AGR 0 time :

rref_AGR A rref : 10000 0.5 - 0.5 11 - 0.5 0.5 - 0.5 0.5 0.5 - 01000 1.5 0.5 11 0.5 0.5 - 0.5 0.5 - 1.5 00100 0.5 - 0.5 - 00 0.5 0.5 - 0.5 0.5 0.5 00010 1.5 0.5 01 0.5 0.5 0.5 0.5 - 0.5 00001 0.5 - 0.5 - 00 0.5 - 0.5 0.5 - 0.5 0.5 - 514 mat

rref_AGR A rref : 10 0.1667 0011 - 1 - 000034 mat

rref_AGR A rref : 10 20.5 01 2.5 - 00000043 mat

rref_AGR A rref : 0100000100000100000000000000000000075 mat

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

toc_AGR 0 time :

Δt_Maple toc_Maple tic_Maple - : ms 116

Δt_Maple toc_Maple tic_Maple - : ms 101

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_Maple toc_Maple tic_Maple - : ms 98

Δt_AGR toc_AGR tic_AGR - : ms 22

Δt_AGR toc_AGR tic_AGR - : ms 15

Δt_AGR toc_AGR tic_AGR - : ms 0

Δt_AGR toc_AGR tic_AGR - : ms 31

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

rref_AGR rref_Maple ≡ 1

ROUND 10 - BIG MATRICES

One disadvantage I have found is that you have to evaluate (show) the matrix if you want to use Maple rref(). This is an inconvenient when you have matrices generated inside functions and you cannot display them.

Range for numbers [0 0.6] generates very good examples of singularity test.

lb 100 - :

ub 100 :

m 50 random : 29

n 50 random : 35

A lb ub 12 mat m n randi : 43 - 11 mat

tic_Maple 0 time :

rref_Maple A rref maple : 10000000000000000000000000000 0.1055 0.2597 - 3.1003 - 1.3168 0.7398 0.8109 01000000000000000000000000000 0.0559 - 0.1773 0.2033 - 0.0799 0.2896 - 0.0207 00100000000000000000000000000 1.6669 - 0.6669 14.2381 - 3.1768 2.4888 - 0.2419 - 00010000000000000000000000000 0.7543 - 1.2936 - 5.0166 - 3.1367 0.2722 0.6218 - 00001000000000000000000000000 0.4543 0.129 6.2558 0.3824 - 1.5018 0.9026 - 00000100000000000000000000000 0.1756 0.1291 4.8381 0.8426 - 0.1302 0.2953 00000010000000000000000000000 1.5394 - 0.1308 2.2182 - 1.181 0.7487 - 1.4185 - 00000001000000000000000000000 1.1328 0.3497 1.4774 1.3259 - 0.2008 0.7747 00000000100000000000000000000 0.786 0.02 - 4.801 - 2.0799 - 0.0884 - 0.9421 00000000010000000000000000000 0.0203 0.4957 0.6522 - 0.7707 0.767 0.7154 00000000001000000000000000000 0.3874 - 0.3599 4.0895 - 1.2597 0.5913 - 0.2526 - 00000000000100000000000000000 0.3416 0.4525 7.8351 1.6782 - 0.3754 0.2231 - 00000000000010000000000000000 0.058 0.248 10.6657 1.9479 - 0.0204 0.7968 - 00000000000001000000000000000 0.0071 - 0.4193 - 9.6291 0.3278 1.7656 0.2419 00000000000000100000000000000 0.3736 - 0.2387 0.4103 - 0.7072 0.6774 - 0.0396 00000000000000010000000000000 1.2853 0.6108 1.4008 - 2.6365 - 0.7361 - 1.4984 00000000000000001000000000000 0.7351 - 0.0486 - 7.9205 - 3.028 0.147 0.1437 00000000000000000100000000000 1.8256 0.2323 4.8492 3.4348 - 0.6119 0.8415 00000000000000000010000000000 1.3247 0.0208 4.5096 - 1.3479 - 0.0987 - 1.0087 00000000000000000001000000000 0.1149 0.437 - 1.5344 - 1.1804 - 0.2707 0.351 00000000000000000000100000000 0.8774 0.3089 7.0228 1.8853 - 0.3013 0.3933 00000000000000000000010000000 0.6994 - 0.8246 4.3118 - 0.5619 1.0778 - 0.0254 00000000000000000000001000000 1.2235 - 0.1331 3.993 - 1.6053 0.6735 - 0.4288 - 00000000000000000000000100000 0.3714 0.2724 - 0.5105 - 0.6858 0.9263 0.3502 00000000000000000000000010000 0.4287 - 0.3595 0.9894 - 1.0218 - 0.8651 - 0.4014 00000000000000000000000001000 0.1973 - 0.4802 - 10.2743 0.0692 1.9885 1.1299 - 00000000000000000000000000100 0.2701 - 0.3861 0.5397 - 0.2712 0.6325 0.3813 - 00000000000000000000000000010 0.624 - 0.5932 1.6017 - 0.7188 1.1616 - 0.5095 - 00000000000000000000000000001 0.5652 - 0.0341 - 5.1334 - 1.4808 0.971 - 0.3867 - 2935 mat

toc_Maple 0 time :

tic_AGR 0 time :

rref_AGR A rref : 10000000000000000000000000000 0.1055 0.2597 - 3.1003 - 1.3168 0.7398 0.8109 01000000000000000000000000000 0.0559 - 0.1773 0.2033 - 0.0799 0.2896 - 0.0207 00100000000000000000000000000 1.6669 - 0.6669 14.2381 - 3.1768 2.4888 - 0.2419 - 00010000000000000000000000000 0.7543 - 1.2936 - 5.0166 - 3.1367 0.2722 0.6218 - 00001000000000000000000000000 0.4543 0.129 6.2558 0.3824 - 1.5018 0.9026 - 00000100000000000000000000000 0.1756 0.1291 4.8381 0.8426 - 0.1302 0.2953 00000010000000000000000000000 1.5394 - 0.1308 2.2182 - 1.181 0.7487 - 1.4185 - 00000001000000000000000000000 1.1328 0.3497 1.4774 1.3259 - 0.2008 0.7747 00000000100000000000000000000 0.786 0.02 - 4.801 - 2.0799 - 0.0884 - 0.9421 00000000010000000000000000000 0.0203 0.4957 0.6522 - 0.7707 0.767 0.7154 00000000001000000000000000000 0.3874 - 0.3599 4.0895 - 1.2597 0.5913 - 0.2526 - 00000000000100000000000000000 0.3416 0.4525 7.8351 1.6782 - 0.3754 0.2231 - 00000000000010000000000000000 0.058 0.248 10.6657 1.9479 - 0.0204 0.7968 - 00000000000001000000000000000 0.0071 - 0.4193 - 9.6291 0.3278 1.7656 0.2419 00000000000000100000000000000 0.3736 - 0.2387 0.4103 - 0.7072 0.6774 - 0.0396 00000000000000010000000000000 1.2853 0.6108 1.4008 - 2.6365 - 0.7361 - 1.4984 00000000000000001000000000000 0.7351 - 0.0486 - 7.9205 - 3.028 0.147 0.1437 00000000000000000100000000000 1.8256 0.2323 4.8492 3.4348 - 0.6119 0.8415 00000000000000000010000000000 1.3247 0.0208 4.5096 - 1.3479 - 0.0987 - 1.0087 00000000000000000001000000000 0.1149 0.437 - 1.5344 - 1.1804 - 0.2707 0.351 00000000000000000000100000000 0.8774 0.3089 7.0228 1.8853 - 0.3013 0.3933 00000000000000000000010000000 0.6994 - 0.8246 4.3118 - 0.5619 1.0778 - 0.0254 00000000000000000000001000000 1.2235 - 0.1331 3.993 - 1.6053 0.6735 - 0.4288 - 00000000000000000000000100000 0.3714 0.2724 - 0.5105 - 0.6858 0.9263 0.3502 00000000000000000000000010000 0.4287 - 0.3595 0.9894 - 1.0218 - 0.8651 - 0.4014 00000000000000000000000001000 0.1973 - 0.4802 - 10.2743 0.0692 1.9885 1.1299 - 00000000000000000000000000100 0.2701 - 0.3861 0.5397 - 0.2712 0.6325 0.3813 - 00000000000000000000000000010 0.624 - 0.5932 1.6017 - 0.7188 1.1616 - 0.5095 - 00000000000000000000000000001 0.5652 - 0.0341 - 5.1334 - 1.4808 0.971 - 0.3867 - 2935 mat

toc_AGR 0 time :

Δt_Maple toc_Maple tic_Maple - : ms 4108

Δt_AGR toc_AGR tic_AGR - : ms 1703

rref_AGR rref_Maple ≡ 0

REVIEW RREF DUEL - Matrix size: m eval num2str x n eval num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 29 x 7, with integer numbers from 0 to50. rref_MAPLE calculation time was: 1166 ms. rref_AGR calculation time was: 497 ms. Maximum absolute error between elements AGR and MAPLE: 1.15E-13 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 26 x 10, with integer numbers from 0 to50. rref_MAPLE calculation time was: 169 ms. rref_AGR calculation time was: 116 ms. Maximum absolute error between elements AGR and MAPLE: 0 41 mat

REVIEW RREF DUEL - Matrix size: m eval num2str x n eval num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 45 x 21, with integer numbers from 0 to50. rref_MAPLE calculation time was: 926 ms. rref_AGR calculation time was: 1077 ms. Maximum absolute error between elements AGR and MAPLE: 0 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 43 x 23, with integer numbers from 0 to50. rref_MAPLE calculation time was: 1114 ms. rref_AGR calculation time was: 1166 ms. Maximum absolute error between elements AGR and MAPLE: 0 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 32 x 35, with integer numbers from 0 to50. rref_MAPLE calculation time was: 2424 ms. rref_AGR calculation time was: 1908 ms. Maximum absolute error between elements AGR and MAPLE: 3.23827E-10 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 26 x 28, with integer numbers from 0 to1. rref_MAPLE calculation time was: 670 ms. rref_AGR calculation time was: 472 ms. Maximum absolute error between elements AGR and MAPLE: 0 41 mat

S

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 24 x 30, with integer numbers from 0 to50. rref_MAPLE calculation time was: 1776 ms. rref_AGR calculation time was: 962 ms. Maximum absolute error between elements AGR and MAPLE: 2.816E-12 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub var2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 23 x 32, with integer numbers from 0 to 0.6. rref_MAPLE calculation time was: 853 ms. rref_AGR calculation time was: 487 ms. Maximum absolute error between elements AGR and MAPLE: 0 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub num2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 38 x 34, with integer numbers from 0 to50. rref_MAPLE calculation time was: 1700 ms. rref_AGR calculation time was: 2275 ms. Maximum absolute error between elements AGR and MAPLE: 0 41 mat

REVIEW RREF DUEL - Matrix size: m num2str x n num2str, with integer numbers from lb num2str to ub var2str . concat rref_MAPLE calculation time was: Δt_Maple ms / eval num2str ms. concat rref_AGR calculation time was: Δt_AGR ms / eval num2str ms. concat Maximum absolute error between elements AGR and MAPLE: rref_Maple rref_AGR - abs vectorize max eval var2str concat 41 mat : RREF DUEL - Matrix size: 29 x 35, with integer numbers from -100 to 100. rref_MAPLE calculation time was: 4108 ms. rref_AGR calculation time was: 1703 ms. Maximum absolute error between elements AGR and MAPLE: 2.1485E-11 41 mat