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# Partial fraction expansion # # Example # # expand(1/(x^3+x^2),x) # # 1 1 1 # ---- - --- + ------- # 2 x x + 1 # x Eval_expand = -> # 1st arg push(cadr(p1)) Eval() # 2nd arg push(caddr(p1)) Eval() p2 = pop() if (p2 == symbol(NIL)) guess() else push(p2) expand() #define A p2 #define B p3 #define C p4 #define F p5 #define P p6 #define Q p7 #define T p8 #define X p9 expand = -> save() p9 = pop() p5 = pop() if (istensor(p5)) expand_tensor() restore() return # if sum of terms then sum over the expansion of each term if (car(p5) == symbol(ADD)) push_integer(0) p1 = cdr(p5) while (iscons(p1)) push(car(p1)) push(p9) expand() add() p1 = cdr(p1) restore() return # B = numerator push(p5) numerator() p3 = pop() # A = denominator push(p5) denominator() p2 = pop() remove_negative_exponents() # Q = quotient push(p3) push(p2) push(p9) divpoly() p7 = pop() # remainder B = B - A * Q push(p3) push(p2) push(p7) multiply() subtract() p3 = pop() # if the remainder is zero then we're done if (iszero(p3)) push(p7) restore() return # A = factor(A) push(p2) push(p9) factorpoly() p2 = pop() expand_get_C() expand_get_B() expand_get_A() if (istensor(p4)) push(p4) inv() push(p3) inner() push(p2) inner() else push(p3) push(p4) divide() push(p2) multiply() push(p7) add() restore() expand_tensor = -> i = 0 push(p5) copy_tensor() p5 = pop() for i in [0...p5.tensor.nelem] push(p5.tensor.elem[i]) push(p9) expand() p5.tensor.elem[i] = pop() push(p5) remove_negative_exponents = -> h = 0 i = 0 j = 0 k = 0 n = 0 h = tos factors(p2) factors(p3) n = tos - h # find the smallest exponent j = 0 for i in [0...n] p1 = stack[h + i] if (car(p1) != symbol(POWER)) continue if (cadr(p1) != p9) continue push(caddr(p1)) k = pop_integer() if (k == 0x80000000) continue if (k < j) j = k tos = h if (j == 0) return # A = A / X^j push(p2) push(p9) push_integer(-j) power() multiply() p2 = pop() # B = B / X^j push(p3) push(p9) push_integer(-j) power() multiply() p3 = pop() # Returns the expansion coefficient matrix C. # # Example: # # B 1 # --- = ----------- # A 2 # x (x + 1) # # We have # # B Y1 Y2 Y3 # --- = ---- + ---- + ------- # A 2 x x + 1 # x # # Our task is to solve for the unknowns Y1, Y2, and Y3. # # Multiplying both sides by A yields # # AY1 AY2 AY3 # B = ----- + ----- + ------- # 2 x x + 1 # x # # Let # # A A A # W1 = ---- W2 = --- W3 = ------- # 2 x x + 1 # x # # Then the coefficient matrix C is # # coeff(W1,x,0) coeff(W2,x,0) coeff(W3,x,0) # # C = coeff(W1,x,1) coeff(W2,x,1) coeff(W3,x,1) # # coeff(W1,x,2) coeff(W2,x,2) coeff(W3,x,2) # # It follows that # # coeff(B,x,0) Y1 # # coeff(B,x,1) = C Y2 # # coeff(B,x,2) = Y3 # # Hence # # Y1 coeff(B,x,0) # -1 # Y2 = C coeff(B,x,1) # # Y3 coeff(B,x,2) expand_get_C = -> h = 0 i = 0 j = 0 n = 0 #U **a h = tos if (car(p2) == symbol(MULTIPLY)) p1 = cdr(p2) while (iscons(p1)) p5 = car(p1) expand_get_CF() p1 = cdr(p1) else p5 = p2 expand_get_CF() n = tos - h if (n == 1) p4 = pop() return p4 = alloc_tensor(n * n) p4.tensor.ndim = 2 p4.tensor.dim[0] = n p4.tensor.dim[1] = n a = h for i in [0...n] for j in [0...n] push(stack[a+j]) push(p9) push_integer(i) power() divide() push(p9) filter() p4.tensor.elem[n * i + j] = pop() tos -= n # The following table shows the push order for simple roots, repeated roots, # and inrreducible factors. # # Factor F Push 1st Push 2nd Push 3rd Push 4th # # # A # x --- # x # # # 2 A A # x ---- --- # 2 x # x # # # A # x + 1 ------- # x + 1 # # # 2 A A # (x + 1) ---------- ------- # 2 x + 1 # (x + 1) # # # 2 A Ax # x + x + 1 ------------ ------------ # 2 2 # x + x + 1 x + x + 1 # # # 2 2 A Ax A Ax # (x + x + 1) --------------- --------------- ------------ ------------ # 2 2 2 2 2 2 # (x + x + 1) (x + x + 1) x + x + 1 x + x + 1 # # # For T = A/F and F = P^N we have # # # Factor F Push 1st Push 2nd Push 3rd Push 4th # # x T # # 2 # x T TP # # # x + 1 T # # 2 # (x + 1) T TP # # 2 # x + x + 1 T TX # # 2 2 # (x + x + 1) T TX TP TPX # # # Hence we want to push in the order # # T * (P ^ i) * (X ^ j) # # for all i, j such that # # i = 0, 1, ..., N - 1 # # j = 0, 1, ..., deg(P) - 1 # # where index j runs first. expand_get_CF = -> d = 0 i = 0 j = 0 n = 0 if (!Find(p5, p9)) return trivial_divide() if (car(p5) == symbol(POWER)) push(caddr(p5)) n = pop_integer() p6 = cadr(p5) else n = 1 p6 = p5 push(p6) push(p9) degree() d = pop_integer() for i in [0...n] for j in [0...d] push(p8) push(p6) push_integer(i) power() multiply() push(p9) push_integer(j) power() multiply() # Returns T = A/F where F is a factor of A. trivial_divide = -> h = 0 if (car(p2) == symbol(MULTIPLY)) h = tos p0 = cdr(p2) while (iscons(p0)) if (!equal(car(p0), p5)) push(car(p0)) Eval(); # force expansion of (x+1)^2, f.e. p0 = cdr(p0) multiply_all(tos - h) else push_integer(1) p8 = pop() # Returns the expansion coefficient vector B. expand_get_B = -> i = 0 n = 0 if (!istensor(p4)) return n = p4.tensor.dim[0] p8 = alloc_tensor(n) p8.tensor.ndim = 1 p8.tensor.dim[0] = n for i in [0...n] push(p3) push(p9) push_integer(i) power() divide() push(p9) filter() p8.tensor.elem[i] = pop() p3 = p8 # Returns the expansion fractions in A. expand_get_A = -> h = 0 i = 0 n = 0 if (!istensor(p4)) push(p2) reciprocate() p2 = pop() return h = tos if (car(p2) == symbol(MULTIPLY)) p8 = cdr(p2) while (iscons(p8)) p5 = car(p8) expand_get_AF() p8 = cdr(p8) else p5 = p2 expand_get_AF() n = tos - h p8 = alloc_tensor(n) p8.tensor.ndim = 1 p8.tensor.dim[0] = n for i in [0...n] p8.tensor.elem[i] = stack[h + i] tos = h p2 = p8 expand_get_AF = -> d = 0 i = 0 j = 0 n = 1 if (!Find(p5, p9)) return if (car(p5) == symbol(POWER)) push(caddr(p5)) n = pop_integer() p5 = cadr(p5) push(p5) push(p9) degree() d = pop_integer() for i in [n...0] for j in [0...d] push(p5) push_integer(i) power() reciprocate() push(p9) push_integer(j) power() multiply()