dd63489a-78ad-4cc5-86f8-09b40e0e6e45 45 2 5 false false 1 decimal

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

Echelon form of a given matrix_M echel ech_M : np 1 : gen 1 ech cols range iz 0 ech cols matrix : inz 0 ech cols matrix : p np : j 1 ech rows range ech j gen el 0 ≡ (j p ≥ (p 1 > (* 1 p 1 ≡ (* + (& iz iz ech j row stack : 11 line inz inz ech j row stack : 11 line if 11 line for ech inz iz stack : pr ech p row : pr gen el 0 ≠ prm pr pr gen el / : np p 1 + ech rows 12 mat min : 21 line prm pr : np p : 21 line if M1 p 1 > ech 1 p 1 - 1 ech cols submatrix 0 ech cols matrix if : M2 prm : temp 0 ech cols matrix : M3 p ech rows ≡ 40411 mat 11 line j p 1 + (ech rows range temp temp ech j row ech j gen el prm * - stack : for temp 21 line if : ech M3 40411 mat ≡ M1 M2 stack M1 M2 M3 stack if : prm gen el 1 ≡ (p ech rows ≡ (& break continue if 131 line for ech 41 line :

Row Reduced Echelon Form (RREF)_M rref_eM_M echel : pivot 0_M cols matrix : j 1_eM rows range_eM j row 1_eM cols matrix ≠ pivot pivot_eM j row stack : 11 line continue 11 line if 11 line for temp pivot reverse : j 1 temp rows range k 1 temp cols range temp j k el 0 ≠ pos j el k : break 21 line continue 11 line if 11 line for 11 line for result_p temp 1 row : temp rows 1 > j 2 temp rows range row_mod temp j row : k 1 j 1 - (range sust_p temp j pos k el el result_p k row * : row_mod row_mod sust_p - : 21 line for result_p result_p row_mod stack : 31 line for 11 line result_p result_p : 11 line if result_f result_p reverse_eM rows temp rows -_eM cols matrix stack : 81 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 : 010000000000000000001010000000000011000058 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 : 010000000011000000001010000000000000000058 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 : 010000000011000000001010000000000000000058 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 90

Δt_Maple toc_Maple tic_Maple - : ms 110

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_Maple toc_Maple tic_Maple - : ms 100

Δt_Maple toc_Maple tic_Maple - : ms 96

Δt_Maple toc_Maple tic_Maple - : ms 110

Δt_AGR toc_AGR tic_AGR - : ms 35

Δt_AGR toc_AGR tic_AGR - : ms 11

Δt_AGR toc_AGR tic_AGR - : ms 10

Δt_AGR toc_AGR tic_AGR - : ms 10

Δt_AGR toc_AGR tic_AGR - : ms 9

Δt_AGR toc_AGR tic_AGR - : ms 30

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 99

Δt_Maple toc_Maple tic_Maple - : ms 90

Δt_Maple toc_Maple tic_Maple - : ms 99

Δt_Maple toc_Maple tic_Maple - : ms 90

Δt_Maple toc_Maple tic_Maple - : ms 87

Δt_AGR toc_AGR tic_AGR - : ms 101

Δt_AGR toc_AGR tic_AGR - : ms 20

Δt_AGR toc_AGR tic_AGR - : ms 40

Δt_AGR toc_AGR tic_AGR - : ms 199

Δt_AGR toc_AGR tic_AGR - : ms 13

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 111

Δt_Maple toc_Maple tic_Maple - : ms 90

Δt_Maple toc_Maple tic_Maple - : ms 90

Δt_Maple toc_Maple tic_Maple - : ms 94

Δt_AGR toc_AGR tic_AGR - : ms 59

Δt_AGR toc_AGR tic_AGR - : ms 21

Δt_AGR toc_AGR tic_AGR - : ms 9

Δt_AGR toc_AGR tic_AGR - : ms 26

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.

A 0 0.6 12 mat 25 random 25 random randi : 001001000000000000000000100000000000010100000100000000010010000101000000100100000100000010910 mat

tic_Maple 0 time :

rref_Maple A rref maple : 01000000000000101 - 01 - 0000001000000000001 - 00000000000100000000001 - 00000000000010000000000000000000000010000000010000000000000001011000000000000000000001000000010100000000000000000000000000000000000000000000000000923 mat

toc_Maple 0 time :

rref_AGR A rref : 01000000000000101 - 01 - 0000001000000000001 - 00000000000100000000001 - 00000000000010000000000000000000000010000000010000000000000001011000000000000000000001000000010100000000000000000000000000000000000000000000000000923 mat

tic_AGR 0 time :

toc_AGR 0 time :

Δt_Maple toc_Maple tic_Maple - : ms 150

Δt_AGR toc_AGR tic_AGR - : ms 10

rref_AGR rref_Maple ≡ 1