c07c2d44-4894-41f5-9b66-e111480473b5 22 3 5 decimal

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

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 0.000000000000031 0.00000000000003 - 0.00000000000002 - 31 mat

x y z f 5 x * 2 y * + z + 36 ≡ x 7 y * + 3 z * + 63 ≡ 2 x * 3 y * + 8 z * + 81 ≡ 31 mat :

x y z f x y z 31 mat roots 3.6 5.4 7.2 31 mat

user defined 'solve' f v Solve f v roots :

xx yy zz f 5 xx * 2 yy * + zz + 36 ≡ xx 7 yy * + 3 zz * + 63 ≡ 2 xx * 3 yy * + 8 zz * + 81 ≡ 31 mat :

xx yy zz f xx yy zz 31 mat Solve 3.6 5.4 7.2 31 mat

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

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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

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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 [w,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 !

[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

Solving non-algebraic wrt z²

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 : 1.42246939034512 0.975388528087757 0.768008037465102 31 mat

MCD

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

m1 1 - :

x y eq1 m1 x * y + :

m2 5.1446 :

R π :

x y eq2 m2 x R - (* y + :

eq1 eq2 2.63 2.63 2.63 2.63 2 2 mat 3 1 sys

x y q x y eq1 x y eq2 21 mat :

sol x y q x y 21 mat roots : 2.63 2.63 21 mat

x y sol x y 21 sys x y q x y 21 mat roots at :

sol 2.6303156536891 2.6303156536891 21 mat

x y sol x y 21 sys eq1 eq2 21 sys x y 21 sys solve at :

Algebraic 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 0.000000000000031 0.00000000000003 - 0.00000000000002 - 31 mat

Solve explicit

m1 1 - :

R π :

x y eq1 m1 x sqrt * y + :

m2 5.1446 :

x y eq2 m2 x R - (* y + :

x y m1 - (x sqrt * : x u m2 - (x R - (* : 21 sys

X x y x u - 0 ≡ x 2.5 3 solve : Y X u : 21 sys

x y x u X Y . 14 black 1 5 mat 3 1 sys

X Y 12 mat 2.815 1.678 12 mat