730a3d90-1db2-4617-b448-586f56550fd4 49 15 12 decimal

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

Given an [x,y,z] system in book style .......... ......................convert to compactum style

x 1 : y 1 : z 1 : 13 mat

5 x * 2 y * + z + x 7 y * 3 z * + 2 x * 3 y * + 8 z * 33 mat 36638131 mat 12 mat

5 x * 2 y * + z + x 7 y * 3 z * + 2 x * 3 y * + 8 z * 33 mat 52117323833 mat

.. plug in the compactum and solve

[x,y,z] solve compactum ........ U 52117323833 mat : V 36638131 mat : xyz U 1 -^V * : 15 mat 21 line

[x,y,z] solutions, sanity xyz U xyz * 12 mat 3.6 5.4 7.2 31 mat 36638131 mat 12 mat

clear the compactum for re-use U U Clear : V V Clear : 12 mat [x,y,z] solve compactum ........ U 107877565861097591044 mat : V 3223333141 mat : wxyz U 1 -^V * : 15 mat 41 line

[w,x,y,z] solutions and sanity wxyz U wxyz * 12 mat 111141 mat 3223333141 mat 12 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 !

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 :

wxyz U 1 -^V * :

<= So much from so little !

[w,x,y,z] ... solutions and sanity wxyz U wxyz * 12 mat 121.169 - 203.704 51.24 - 32.329 41 mat 3223333141 mat 12 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

I do hate sums. There is no greater mistake than to call arithmetic an exact sience. There are ... hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different. Mrs> La Touche [19th Century]

Transformation is not "affine" wrt the ratio of the lengths

To make it worst: Not exactly "1".

a 82^32^+ sqrt 52^22^+ sqrt / :

b 52^22^+ sqrt 82^32^+ sqrt / :

a 2^b 2^+ sqrt 112^12^+ sqrt / - 1.000084

a 1.586582

b 0.630286

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

a b f 3 a * 4 b * - 24 ≡ 5 a * 2 b * + 3 - ≡ 21 mat :

5346 does not support syntax a b 21 mat a b f a b 21 mat roots : 1.384615 4.961538 - 21 mat

a b 21 mat 1.3846 4.9615 - 21 mat

alternately: collect solutions in vector r a b f a b 21 mat roots : 1.384615 4.961538 - 21 mat

r sol r 1 el r 2 el f :

15 D numeric
exp threshold  15 r sol 0021 mat

Sanity check

15 D, 15 exp threshold 6923076923076950000000000000 / 496153846153847100000000000000 / - f 0021 mat

36923076923076950000000000000 / * 4496153846153847100000000000000 / - (* - 24

56923076923076950000000000000 / * 2496153846153847100000000000000 / - (* + 3.00000000000004 -

This is not "sanity". Different numerical context => results in different numerical issues. Mathematica would explain: different results may result from different ULP in the execution context. ULP = "Unit in the Last Place"

r sol 3410605131648470142108547152020 / 24 ≡ 1091393642127510363797880709171 / - 3 - ≡ 21 mat

24436557456851006001818989403545860 / - 2.199023 1014 -^*

3 - (1091393642127530363797880709171 / - (- 4.672924 1014 -^*

r 1.38461538461538461538 4.96153846153846153846 - 21 mat :

3 r 1 el * 4 r 2 el (* - 24

5 r 1 el * 2 r 2 el (* + 3.00000000000002 -

A 100100 2.7 2.5 1000 2.66 - 5.2 5.2 044 mat :

C 104002704002704041 mat :

A 1 -^C * 0 4814.814815 385.185185 1040041 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 00031 mat

Implicit solve ... roots

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

sol x y z 621 f x y z 31 mat roots : 1.818182 2.363636 2.181818 - 31 mat