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tensor design
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;; | |
;; Tensor | |
;; | |
[| x |] | |
;=> | |
x | |
(+ 1 [|1 2 3|]) | |
;=> | |
[|2 3 4|] | |
(+ [|1 2 3|] 1) | |
;=> | |
[|2 3 4|] | |
(∂/∂ [|(f x) (g x)|] x) | |
;=> | |
[|(f_1 x) (g_1 x)|] | |
(+ [|1 2 3|] [|1 2 3|]) | |
;=> | |
[|2 4 6|] | |
(+ [|1 2 3|]_i | |
[|[|1 2 3|] [|10 20 30|]|]_i_j) | |
;=> | |
[|[|2 4 6|] [|11 22 33|]|]_i_j | |
(apply [|(∂/∂ $ x) (∂/∂ $ y)|] (f x y)) | |
;=> | |
[|(f_1 x y) (f_2 x y)|] | |
;; | |
;; Derivative | |
;; | |
(f x y z) | |
(∂/∂ (f x y z) x) | |
;=> | |
(f_1 x y z) | |
(∂/∂ (f x y z) [|x y z|]) | |
;=> | |
[|(f_1 x y z) (f_2 x y z) (f_3 x y z)|] | |
;; | |
;; Nabla | |
;; | |
(define $∇ ∂/∂) | |
(∇ (f x y z)) [|x y z|]) | |
;=> | |
[|(f_1 x y z) (f_2 x y z) (f_3 x y z)|] | |
(∇ [|(f1 x y z) (f2 x y z) (f3 x y z)|] [|x y z|]) | |
;=> | |
[|[|(f1_1 x y z) (f1_2 x y z) (f1_3 x y z)|] | |
[|(f2_1 x y z) (f2_2 x y z) (f2_3 x y z)|] | |
[|(f3_1 x y z) (f3_2 x y z) (f3_3 x y z)|] | |
|] | |
;; | |
;; Divergence | |
;; | |
(define $div (compose ∇ trace)) | |
(div [|(f x y z) (g x y z) (h x y z)|] [|x y z|]) | |
;=> | |
(+ (f_1 x y z) (g_2 x y z) (h_3 x y z)) | |
;; | |
;; Taylor Expansion | |
;; | |
(define $multivariate-taylor-expansion | |
(lambda [|$f $xs $as|] | |
(with-symbols {h} | |
(let {[|$hs (generate-tensor 1#h_%1 (tensor-size xs))|]} | |
(map2 * | |
(map 1#(/ 1 (fact %1)) nats0) | |
(map (compose (substitute xs as $) | |
(substitute hs (- xs as) $)) | |
(iterate (compose (∇ $ xs) (V.* hs $)) f))))))) | |
(take 3 (multivariate-taylor-expansion (f x y) [|x y|] [|0 0|])) | |
;=> | |
{(f 0 0) | |
(+ (* x (f_1 0 0)) | |
(* y (f_2 0 0))) | |
(/ (+ (* x^2 (f_1_1 0 0)) | |
(* x y (f_2_1 0 0)) | |
(* y x (f_1_2 0 0)) | |
(* y^2 (f_2_2 0 0))) | |
2)} |
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