47280413-d50b-428a-8830-4e48853de5a1 8 15 12 decimal

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

# # # if

Solve implicit algebraic

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

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

sanity x y z f 0.000000000000028 0.000000000000031 - 0.00000000000002 - 31 mat

x y z Clear 1

Solve implicit NON-algebraic . . .

A difficult task as many sets of solutions are expected.

In this example, we get [3] sets quite easily. No known

rule(s) are suggested ... trial/error.

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Non-algebraic system f 2 x * y + 2 z 2^* + 5 - 0 ≡ y 3^4 z * + 4 - 0 ≡ x y * z + e z^- 0 ≡ 31 mat :

seed init x 1 ≡ y 1 ≡ z 0.1 ≡ 31 mat :

Accuracy ε 1014 -^:

x y z 31 mat f init ε FindRoot : 1.42246939034512 0.975388528087757 0.768008037465102 31 mat

MCD single set

[set: 1]

sanity 2 x * y + 2 z 2^* + 5 - y 3^4 z * + 4 - x y * z + e z^- 31 mat : 71015 -^* - 11015 -^* - 11014 -^* - 31 mat

x y z Clear 1

Non-algebraic system f 2 x * y + 2 z 2^* + 5 - 0 ≡ y 3^4 z * + 4 - 0 ≡ x y * z + e z^- 0 ≡ 31 mat :

seed init x 1 ≡ y 1 ≡ z 0.1 - ≡ 31 mat :

Accuracy ε 1014 -^:

x y z 31 mat f init ε FindRoot : 1.41584424915365 0.976877013601305 0.777869429079966 31 mat

[set: 2]

sanity 2 x * y + 2 z 2^* + 5 - y 3^4 z * + 4 - x y * z + e z^- 31 mat : 00031 mat

x y z Clear 1

Non-algebraic system f 2 x * y + 2 z 2^* + 5 - 0 ≡ y 3^4 z * + 4 - 0 ≡ x y * z + e z^- 0 ≡ 31 mat :

seed init x 1 ≡ y 1 - ≡ z 1 ≡ 31 mat :

Accuracy ε 1014 -^:

x y z 31 mat f init ε FindRoot : 1.42300421336094 0.974992590477666 0.768057425666393 31 mat

[set: 3]

sanity 2 x * y + 2 z 2^* + 5 - y 3^4 z * + 4 - x y * z + e z^- 31 mat : 00031 mat

M => tridiagonal matrix of the "linear cubic spline"
'b' => the vector of derivatives
s => vector coefficients of the linear cubic spline M 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat :

b 0 0.6782 1.44 - 0.74 0.052 - 061 mat :

s M 1 -^b * : 0 2.4192 3.2851 - 2.0809 0.5983 - 061 mat

[x,y,z] solve compactum ........ A 03534 - 050633 mat : b 1.20736 2.34066 - 0.329193 - 31 mat : A b Jacobi A A 1 -^b * (* : 15 mat 21 line

u A 1 -^b * : 0.142856087 0.692307065 0.173912239 - 31 mat

A u * 1.20736 2.34066 - 0.329193 - 31 mat

A b Jacobi 1.20736 2.34066 - 0.329193 - 31 mat

x A u * 1 el : y A u * 2 el : z A u * 3 el : 31 mat 1.20736 2.34066 - 0.329193 - 31 mat

==========================================================

clear the compactum for re-use U U Clear : V V Clear : 12 mat [w,x,y,z] 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 41 line

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

Use the compactum wisely, rename the variables [w,x,y,z] solve compactum ........ u 107 8.1 7.201 7.08 5.04 658 5.98 9.89 9 6.99 4.99 9 9.98 44 mat : v 3223333141 mat : sol u 1 -^v * : 15 mat 21 line

[w,x,y,z] solutions sol 121.168757 - 203.704132 51.24021 - 32.329448 41 mat

So much from so little !

[w,x,y,z]  solutions/sanity sol u sol * 12 mat 121.168757 - 203.704132 51.24021 - 32.329448 41 mat 3223333141 mat 12 mat

Plug this algo in utilities,
collapsed top document. 
Collected enables elegant
description ... Ah ! U V solve x y z 31 mat U 1 -^V * : 11 line 11 sys :

u 12 - 12 - 41 - 011 - 33 mat :

v 4 - 23 - 31 mat :

solutions are in ...
lexicographic order. x y z 31 mat u v solve : 16 - 9 - 6 - 31 mat

x y z 31 mat 12 - 12 - 41 - 011 - 33 mat 4.56789 - 2.987654321 2 sqrt - 31 mat solve (: 13.5444428037462 - 7.56233924137309 - 6.148125679 - 31 mat

collapse in utilities ... @ top of document m b collect i 1 b cols range collect i el m b i el solve : for collect transpose 21 line :

b 4 - 23 - 31 mat 4.56789 - 2.987654321 2 sqrt - 31 mat 4.56789 2.987654321 e sqrt - 31 mat 13 mat :

m 12 - 12 - 41 - 011 - 33 mat :

sol m b collect (: 16 - 9 - 6 - 31 mat 13.5444428037462 - 7.56233924137309 - 6.148125679 - 31 mat 13.3938817795997 10.4747130502999 12.123434321 31 mat 13 mat

j sanity m sol 1 j el * :

1 sanity 2 sanity 3 sanity 13 mat 4 - 23 - 31 mat 4.56789 - 2.987654321 1.414213562 - 31 mat 4.56789 2.987654321 1.648721271 - 31 mat 13 mat