#define POLY p1 #define X p2 #define A p3 #define B p4 #define C p5 #define Y p6 show_power_debug = false performing_roots = false Eval_roots = -> # A == B -> A - B p2 = cadr(p1) if (car(p2) == symbol(SETQ) || car(p2) == symbol(TESTEQ)) push(cadr(p2)) Eval() push(caddr(p2)) Eval() subtract() else push(p2) Eval() p2 = pop() if (car(p2) == symbol(SETQ) || car(p2) == symbol(TESTEQ)) push(cadr(p2)) Eval() push(caddr(p2)) Eval() subtract() else push(p2) # 2nd arg, x push(caddr(p1)) Eval() p2 = pop() if (p2 == symbol(NIL)) guess() else push(p2) p2 = pop() p1 = pop() if (!ispoly(p1, p2)) stop("roots: 1st argument is not a polynomial") push(p1) push(p2) roots() hasImaginaryCoeff = (k) -> #polycoeff = tos imaginaryCoefficients = false h = tos for i in [k...0] by -1 #console.log "hasImaginaryCoeff - coeff.:" + stack[tos-i].toString() if iscomplexnumber(stack[tos-i]) imaginaryCoefficients = true break return imaginaryCoefficients isSimpleRoot = (k) -> #polycoeff = tos #tos-n Coefficient of x^0 #tos-1 Coefficient of x^(n-1) if k > 2 isSimpleRootPolynomial = true h = tos if iszero(stack[tos-k]) isSimpleRootPolynomial = false for i in [(k-1)...1] by -1 #console.log "hasImaginaryCoeff - coeff.:" + stack[tos-i].toString() if !iszero(stack[tos-i]) isSimpleRootPolynomial = false break else isSimpleRootPolynomial = false return isSimpleRootPolynomial normalisedCoeff = -> k = coeff() #console.log("->" + tos) divideBy = stack[tos-1] miniStack = [] for i in [1..k] miniStack.push(pop()) #console.log(tos) for i in [(k-1)..0] push(miniStack[i]) push(divideBy) divide() #console.log(tos) return k # takes the polynomial and the # variable on the stack roots = -> h = 0 i = 0 n = 0 save() # the simplification of nested radicals uses # "roots", which in turn uses simplification # of nested radicals. Usually there is no problem, # one level of recursion does the job. Beyond that, # we probably got stuck in a strange case of infinite # recursion, so bail out and return NIL. if recursionLevelNestedRadicalsRemoval > 1 pop() pop() push(symbol(NIL)) restore() return performing_roots = true h = tos - 2 if DEBUG then console.log("checking if " + stack[tos-1].toString() + " is a case of simple roots") p2 = pop() p1 = pop() push(p1) push(p2) push(p1) push(p2) k = normalisedCoeff() if isSimpleRoot(k) if DEBUG then console.log("yes, " + stack[tos-1].toString() + " is a case of simple roots") lastCoeff = stack[tos-k] leadingCoeff = stack[tos-1] tos -= k pop() pop() getSimpleRoots(k, leadingCoeff, lastCoeff) else tos -= k roots2() n = tos - h if (n == 0) stop("roots: the polynomial is not factorable, try nroots") if (n == 1) performing_roots = false restore() return sort_stack(n) p1 = alloc_tensor(n) p1.tensor.ndim = 1 p1.tensor.dim[0] = n for i in [0...n] p1.tensor.elem[i] = stack[h + i] tos = h push(p1) restore() performing_roots = false # ok to generate these roots take a look at their form # in the case of even and odd exponents here: # http://www.wolframalpha.com/input/?i=roots+x%5E14+%2B+1 # http://www.wolframalpha.com/input/?i=roots+ax%5E14+%2B+b # http://www.wolframalpha.com/input/?i=roots+x%5E15+%2B+1 # http://www.wolframalpha.com/input/?i=roots+a*x%5E15+%2B+b getSimpleRoots = (n, leadingCoeff, lastCoeff) -> if DEBUG then console.log("getSimpleRoots") save() #tos-n Coefficient of x^0 #tos-1 Coefficient of x^(n-1) n = n-1 push(lastCoeff) push_rational(1,n) power() push(leadingCoeff) push_rational(1,n) power() divide() commonPart = pop() if n % 2 == 0 for rootsOfOne in [1..n] by 2 push(commonPart) push_integer(-1) push_rational(rootsOfOne,n) power() multiply() aSol = pop() push(aSol) push(aSol) negate() else for rootsOfOne in [1..n] push(commonPart) push_integer(-1) push_rational(rootsOfOne,n) power() multiply() if rootsOfOne % 2 == 0 negate() restore() roots2 = -> save() p2 = pop() # the polynomial variable p1 = pop() # the polynomial push(p1) push(p2) push(p1) push(p2) k = normalisedCoeff() if !hasImaginaryCoeff(k) tos -= k factorpoly() p1 = pop() else tos -= k pop() pop() if (car(p1) == symbol(MULTIPLY)) p1 = cdr(p1) # scan through all the factors # and find the roots of each of them while (iscons(p1)) push(car(p1)) push(p2) roots3() p1 = cdr(p1) else push(p1) push(p2) roots3() restore() roots3 = -> save() p2 = pop() p1 = pop() if (car(p1) == symbol(POWER) && ispoly(cadr(p1), p2) && isposint(caddr(p1))) push(cadr(p1)) push(p2) n = normalisedCoeff() mini_solve(n) else if (ispoly(p1, p2)) push(p1) push(p2) n = normalisedCoeff() mini_solve(n) restore() #----------------------------------------------------------------------------- # # Input: stack[tos - 2] polynomial # # stack[tos - 1] dependent symbol # # Output: stack roots on stack # # (input args are popped first) # #----------------------------------------------------------------------------- # note that for many quadratic, cubic and quartic polynomials we don't # actually end up using the quadratic/cubic/quartic formulas in here, # since there is a chance we factored the polynomial and in so # doing we found some solutions and lowered the degree. mini_solve = (n) -> #console.log "mini_solve >>>>>>>>>>>>>>>>>>>>>>>> tos:" + tos save() # AX + B, X = -B/A if (n == 2) #console.log "mini_solve >>>>>>>>> 1st degree" p3 = pop() p4 = pop() push(p4) push(p3) divide() negate() restore() return # AX^2 + BX + C, X = (-B +/- (B^2 - 4AC)^(1/2)) / (2A) if (n == 3) #console.log "mini_solve >>>>>>>>> 2nd degree" p3 = pop() # A p4 = pop() # B p5 = pop() # C # B^2 push(p4) push_integer(2) power() # 4AC push_integer(4) push(p3) multiply() push(p5) multiply() # B^2 - 4AC subtract() #(B^2 - 4AC)^(1/2) push_rational(1, 2) power() #p6 is (B^2 - 4AC)^(1/2) p6 = pop() push(p6); # B push(p4) subtract() # -B + (B^2 - 4AC)^(1/2) # 1/2A push(p3) push_integer(2) multiply() divide() #simplify() #rationalize() # tos - 1 now is 1st root: (-B + (B^2 - 4AC)^(1/2)) / (2A) push(p6); # tos - 1 now is (B^2 - 4AC)^(1/2) # tos - 2: 1st root: (-B + (B^2 - 4AC)^(1/2)) / (2A) # add B to tos push(p4) add() # tos - 1 now is B + (B^2 - 4AC)^(1/2) # tos - 2: 1st root: (-B + (B^2 - 4AC)^(1/2)) / (2A) negate() # tos - 1 now is -B -(B^2 - 4AC)^(1/2) # tos - 2: 1st root: (-B + (B^2 - 4AC)^(1/2)) / (2A) # 1/2A again push(p3) divide() push_rational(1, 2) multiply() #simplify() #rationalize() # tos - 1: 2nd root: (-B - (B^2 - 4AC)^(1/2)) / (2A) # tos - 2: 1st root: (-B + (B^2 - 4AC)^(1/2)) / (2A) restore() return #if (n == 4) if (n == 4 or n == 5) p3 = pop() # A p4 = pop() # B p5 = pop() # C p6 = pop() # D # C - only related calculations push(p5) push(p5) multiply() R_c2 = pop() push(R_c2) push(p5) multiply() R_c3 = pop() # B - only related calculations push(p4) push(p4) multiply() R_b2 = pop() push(R_b2) push(p4) multiply() R_b3 = pop() push(R_b3) push(p6) multiply() R_b3_d = pop() push(R_b3_d) push_integer(-4) multiply() R_m4_b3_d = pop() push(R_b3) push_integer(2) multiply() R_2_b3 = pop() # A - only related calculations push(p3) push(p3) multiply() R_a2 = pop() push(R_a2) push(p3) multiply() R_a3 = pop() push_integer(3) push(p3) multiply() R_3_a = pop() push(R_a2) push(p6) multiply() R_a2_d = pop() push(R_a2_d) push(p6) multiply() R_a2_d2 = pop() push(R_a2_d) push_integer(27) multiply() R_27_a2_d = pop() push(R_a2_d2) push_integer(-27) multiply() R_m27_a2_d2 = pop() push(R_3_a) push_integer(2) multiply() R_6_a = pop() # mixed calculations push(p3) push(p5) multiply() R_a_c = pop() push(R_a_c) push(p4) multiply() R_a_b_c = pop() push(R_a_b_c) push(p6) multiply() R_a_b_c_d = pop() push(R_a_c) push_integer(3) multiply() R_3_a_c = pop() push_integer(-4) push(p3) push(R_c3) multiply() multiply() R_m4_a_c3 = pop() push(R_a_b_c) push_integer(9) multiply() negate() R_m9_a_b_c = pop() push(R_a_b_c_d) push_integer(18) multiply() R_18_a_b_c_d = pop() push(R_b2) push(R_3_a_c) subtract() R_DELTA0 = pop() push(R_b2) push(R_c2) multiply() R_b2_c2 = pop() push(p4) negate() push(R_3_a) divide() R_m_b_over_3a = pop() if (n == 4) if DEBUG then console.log ">>>>>>>>>>>>>>>> actually using cubic formula <<<<<<<<<<<<<<< " #console.log ">>>> A:" + p3.toString() #console.log ">>>> B:" + p4.toString() #console.log ">>>> C:" + p5.toString() #console.log ">>>> D:" + p6.toString() if DEBUG then console.log "cubic: D0: " + R_DELTA0.toString() push(R_DELTA0) push_integer(3) power() push_integer(4) multiply() R_4_DELTA03 = pop() push(R_DELTA0) simplify() absValFloat() R_DELTA0_toBeCheckedIfZero = pop() if DEBUG then console.log "cubic: D0 as float: " + R_DELTA0_toBeCheckedIfZero.toString() #if iszero(R_DELTA0_toBeCheckedIfZero) # console.log " *********************************** D0 IS ZERO" # DETERMINANT push(R_18_a_b_c_d) push(R_m4_b3_d) push(R_b2_c2) push(R_m4_a_c3) push(R_m27_a2_d2) add() add() add() add() simplify() absValFloat() R_determinant = pop() if DEBUG then console.log "cubic: DETERMINANT: " + R_determinant.toString() # R_DELTA1 push(R_2_b3) push(R_m9_a_b_c) push(R_27_a2_d) add() add() R_DELTA1 = pop() if DEBUG then console.log "cubic: D1: " + R_DELTA1.toString() # R_Q push(R_DELTA1) push_integer(2) power() push(R_4_DELTA03) subtract() push_rational(1, 2) power() simplify() R_Q = pop() if iszero(R_determinant) if iszero(R_DELTA0_toBeCheckedIfZero) if DEBUG then console.log " cubic: DETERMINANT IS ZERO and delta0 is zero" push(R_m_b_over_3a) # just same solution three times restore() return else if DEBUG then console.log " cubic: DETERMINANT IS ZERO and delta0 is not zero" push(p3) push(p6) push_integer(9) multiply() multiply() push(p4) push(p5) multiply() subtract() push(R_DELTA0) push_integer(2) multiply() divide() # first solution root_solution = pop() push(root_solution) # pushing two of them on the stack push(root_solution) # second solution here # 4abc push(R_a_b_c) push_integer(4) multiply() # -9a*a*d push(p3) push(p3) push(p6) push_integer(9) multiply() multiply() multiply() negate() # -9*b^3 push(R_b3) negate() # sum the three terms add() add() # denominator is a*delta0 push(p3) push(R_DELTA0) multiply() # build the fraction divide() restore() return C_CHECKED_AS_NOT_ZERO = false flipSignOFQSoCIsNotZero = false # C will go as denominator, we have to check # that is not zero while !C_CHECKED_AS_NOT_ZERO # R_C push(R_Q) if flipSignOFQSoCIsNotZero negate() push(R_DELTA1) add() push_rational(1, 2) multiply() push_rational(1, 3) power() simplify() R_C = pop() if DEBUG then console.log "cubic: C: " + R_C.toString() push(R_C) simplify() absValFloat() R_C_simplified_toCheckIfZero = pop() if DEBUG then console.log "cubic: C as absval and float: " + R_C_simplified_toCheckIfZero.toString() if iszero(R_C_simplified_toCheckIfZero) if DEBUG then console.log " cubic: C IS ZERO flipping the sign" flipSignOFQSoCIsNotZero = true else C_CHECKED_AS_NOT_ZERO = true push(R_C) push(R_3_a) multiply() R_3_a_C = pop() push(R_3_a_C) push_integer(2) multiply() R_6_a_C = pop() # imaginary parts calculations push(imaginaryunit) push_integer(3) push_rational(1, 2) power() multiply() i_sqrt3 = pop() push_integer(1) push(i_sqrt3) add() one_plus_i_sqrt3 = pop() push_integer(1) push(i_sqrt3) subtract() one_minus_i_sqrt3 = pop() push(R_C) push(R_3_a) divide() R_C_over_3a = pop() # first solution push(R_m_b_over_3a) # first term push(R_C_over_3a) negate() # second term push(R_DELTA0) push(R_3_a_C) divide() negate() # third term # now add the three terms together add() add() simplify() # second solution push(R_m_b_over_3a) # first term push(R_C_over_3a) push(one_plus_i_sqrt3) multiply() push_integer(2) divide() # second term push(one_minus_i_sqrt3) push(R_DELTA0) multiply() push(R_6_a_C) divide() # third term # now add the three terms together add() add() simplify() # third solution push(R_m_b_over_3a) # first term push(R_C_over_3a) push(one_minus_i_sqrt3) multiply() push_integer(2) divide() # second term push(one_plus_i_sqrt3) push(R_DELTA0) multiply() push(R_6_a_C) divide() # third term # now add the three terms together add() add() simplify() restore() return # See http://www.sscc.edu/home/jdavidso/Math/Catalog/Polynomials/Fourth/Fourth.html # for a description of general shapes and properties of fourth degree polynomials if (n == 5) if DEBUG then console.log ">>>>>>>>>>>>>>>> actually using quartic formula <<<<<<<<<<<<<<< " p7 = pop() # E if iszero(p4) and iszero(p6) and !iszero(p5) and !iszero(p7) if DEBUG then console.log("biquadratic case") push(p3) push(symbol(SECRETX)) push_integer(2) power() multiply() push(p5) push(symbol(SECRETX)) multiply() push(p7) add() add() push(symbol(SECRETX)) roots() biquadraticSolutions = pop() for eachSolution in biquadraticSolutions.tensor.elem push(eachSolution) push_rational(1,2) power() simplify() push(eachSolution) push_rational(1,2) power() negate() simplify() restore() return # D - only related calculations push(p6) push(p6) multiply() R_d2 = pop() # E - only related calculations push(p7) push(p7) multiply() R_e2 = pop() push(R_e2) push(p7) multiply() R_e3 = pop() # DETERMINANT push_integer(256) push(R_a3) push(R_e3) multiply() multiply() # first term 256 a^3 e^3 push_integer(-192) push(R_a2_d) push(R_e2) push(p4) multiply() multiply() multiply() # second term -192 a^3 b d e^2 push_integer(-128) push(R_a2) push(R_c2) push(R_e2) multiply() multiply() multiply() # third term -128 a^2 c^2 e^2 push_integer(144) push(R_a2_d2) push(p5) push(p7) multiply() multiply() multiply() # fourth term 144 a^2 c d^2 e push(R_m27_a2_d2) push(R_d2) multiply() # fifth term -27 a^2 d^4 push_integer(144) push(R_a_b_c) push(p4) push(R_e2) multiply() multiply() multiply() # sixth term 144 a b^2 c e^2 push_integer(-6) push(p3) push(R_b2) push(R_d2) push(p7) multiply() multiply() multiply() multiply() # seventh term -6 a b^2 d^2 e push_integer(-80) push(R_a_b_c_d) push(p5) push(p7) multiply() multiply() multiply() # eigth term -80 a b c^2 d e push_integer(18) push(R_a_b_c_d) push(R_d2) multiply() multiply() # ninth term 18 a b c d^3 push_integer(16) push(R_a_c) push(R_c3) push(p7) multiply() multiply() multiply() # tenth term 16 a c^4 e push_integer(-4) push(R_a_c) push(R_c2) push(R_d2) multiply() multiply() multiply() # eleventh term -4 a c^3 d^2 push_integer(-27) push(R_b3) push(p4) push(R_e2) multiply() multiply() multiply() # twelveth term -27 b^4 e^2 push_integer(18) push(R_b3_d) push(p5) push(p7) multiply() multiply() multiply() # thirteenth term 18 b^3 c d e push(R_m4_b3_d) push(R_d2) multiply() # fourteenth term -4 b^3 d^3 push_integer(-4) push(R_b2_c2) push(p5) push(p7) multiply() multiply() multiply() # fifteenth term -4 b^2 c^3 e push(R_b2_c2) push(R_d2) multiply() # sixteenth term b^2 c^2 d^2 # add together the sixteen terms by doing # fifteen adds add() add() add() add() add() add() add() add() add() add() add() add() add() add() add() R_determinant = pop() if DEBUG then console.log("R_determinant: " + R_determinant.toString()) # DELTA0 push(R_c2) # term one of DELTA0 push_integer(-3) push(p4) push(p6) multiply() multiply() # term two of DELTA0 push_integer(12) push(p3) push(p7) multiply() multiply() # term three of DELTA0 # add the three terms together add() add() R_DELTA0 = pop() if DEBUG then console.log("R_DELTA0: " + R_DELTA0.toString()) # DELTA1 push_integer(2) push(R_c3) multiply() push_integer(-9) push(p4) push(p5) push(p6) multiply() multiply() multiply() push_integer(27) push(R_b2) push(p7) multiply() multiply() push_integer(27) push(p3) push(R_d2) multiply() multiply() push_integer(-72) push(R_a_c) push(p7) multiply() multiply() # add the five terms together add() add() add() add() R_DELTA1 = pop() if DEBUG then console.log("R_DELTA1: " + R_DELTA1.toString()) # p push_integer(8) push(R_a_c) multiply() push_integer(-3) push(R_b2) multiply() add() push_integer(8) push(R_a2) multiply() divide() R_p = pop() if DEBUG then console.log("p: " + R_p.toString()) # q push(R_b3) push_integer(-4) push(R_a_b_c) multiply() push_integer(8) push(R_a2_d) multiply() add() add() push_integer(8) push(R_a3) multiply() divide() R_q = pop() if DEBUG then console.log("q: " + R_q.toString()) if DEBUG then console.log("tos 1 " + tos) if !iszero(p4) if DEBUG then console.log("tos 2 " + tos) push_integer(8) push(p5) push(p3) multiply() multiply() push_integer(-3) push(p4) push_integer(2) power() multiply() add() push_integer(8) push(p3) push_integer(2) power() multiply() divide() R_p = pop() if DEBUG then console.log("p for depressed quartic: " + R_p.toString()) push(p4) push_integer(3) power() push_integer(-4) push(p3) push(p4) push(p5) multiply() multiply() multiply() push_integer(8) push(p6) push(p3) push_integer(2) power() multiply() multiply() add() add() push_integer(8) push(p3) push_integer(3) power() multiply() divide() R_q = pop() if DEBUG then console.log("q for depressed quartic: " + R_q.toString()) # convert to depressed quartic push(p4) push_integer(4) power() push_integer(-3) multiply() push_integer(256) push(R_a3) push(p7) multiply() multiply() push_integer(-64) push(R_a2_d) push(p4) multiply() multiply() push_integer(16) push(R_b2) push(p3) push(p5) multiply() multiply() multiply() add() add() add() push_integer(256) push(p3) push_integer(4) power() multiply() divide() R_r = pop() if DEBUG then console.log("r for depressed quartic: " + R_r.toString()) if DEBUG then console.log("tos 4 " + tos) push(symbol(SECRETX)) push_integer(4) power() if DEBUG then console.log("4 * x^4: " + stack[tos-1].toString()) push(R_p) push(symbol(SECRETX)) push_integer(2) power() multiply() if DEBUG then console.log("R_p * x^2: " + stack[tos-1].toString()) push(R_q) push(symbol(SECRETX)) multiply() if DEBUG then console.log("R_q * x: " + stack[tos-1].toString()) push(R_r) if DEBUG then console.log("R_r: " + stack[tos-1].toString()) add() add() add() simplify() if DEBUG then console.log("solving depressed quartic: " + stack[tos-1].toString()) push(symbol(SECRETX)) roots() depressedSolutions = pop() if DEBUG then console.log("depressedSolutions: " + depressedSolutions) for eachSolution in depressedSolutions.tensor.elem push(eachSolution) push(p4) push_integer(4) push(p3) multiply() divide() subtract() simplify() if DEBUG then console.log("solution from depressed: " + stack[tos-1].toString()) restore() return else R_p = p5 R_q = p6 R_r = p7 ### # Descartes' solution # https://en.wikipedia.org/wiki/Quartic_function#Descartes.27_solution # finding the "u" in the depressed equation push_integer(2) push(R_p) multiply() coeff2 = pop() push_integer(-4) push(R_p) push_integer(2) power() multiply() push(R_r) multiply() coeff3 = pop() push(R_q) push_integer(2) power() negate() coeff4 = pop() # now build the polynomial push(symbol(SECRETX)) push_integer(3) power() push(coeff2) push(symbol(SECRETX)) push_integer(2) power() multiply() push(coeff3) push(symbol(SECRETX)) multiply() push(coeff4) add() add() add() console.log("Descarte's resolventCubic: " + stack[tos-1].toString()) push(symbol(SECRETX)) roots() resolventCubicSolutions = pop() console.log("Descarte's resolventCubic solutions: " + resolventCubicSolutions) console.log("tos: " + tos) R_u = null #R_u = resolventCubicSolutions.tensor.elem[1] for eachSolution in resolventCubicSolutions.tensor.elem console.log("examining solution: " + eachSolution) push(eachSolution) push_integer(2) multiply() push(R_p) add() absValFloat() toBeCheckedIFZero = pop() console.log("abs value is: " + eachSolution) if !iszero(toBeCheckedIFZero) R_u = eachSolution break console.log("chosen solution: " + R_u) push(R_u) negate() R_s = pop() push(R_p) push(R_u) push_integer(2) power() push(R_q) push(R_u) divide() add() add() push_integer(2) divide() R_t = pop() push(R_p) push(R_u) push_integer(2) power() push(R_q) push(R_u) divide() subtract() add() push_integer(2) divide() R_v = pop() # factoring the quartic into two quadratics: # now build the polynomial push(symbol(SECRETX)) push_integer(2) power() push(R_s) push(symbol(SECRETX)) multiply() push(R_t) add() add() console.log("factored quartic 1: " + stack[tos-1].toString()) push(symbol(SECRETX)) push_integer(2) power() push(R_u) push(symbol(SECRETX)) multiply() push(R_v) add() add() console.log("factored quartic 2: " + stack[tos-1].toString()) pop() restore() return ### # Ferrari's solution # https://en.wikipedia.org/wiki/Quartic_function#Ferrari.27s_solution # finding the "m" in the depressed equation push_rational(5,2) push(R_p) multiply() coeff2 = pop() push_integer(2) push(R_p) push_integer(2) power() multiply() push(R_r) subtract() coeff3 = pop() push(R_p) push_integer(3) power() push_integer(2) divide() push_rational(-1,2) push(R_p) push(R_r) multiply() multiply() push_rational(-1,8) push(R_q) push_integer(2) power() multiply() add() add() coeff4 = pop() push(symbol(SECRETX)) push_integer(3) power() push(coeff2) push(symbol(SECRETX)) push_integer(2) power() multiply() push(coeff3) push(symbol(SECRETX)) multiply() push(coeff4) add() add() add() if DEBUG then console.log("resolventCubic: " + stack[tos-1].toString()) push(symbol(SECRETX)) roots() resolventCubicSolutions = pop() if DEBUG then console.log("resolventCubicSolutions: " + resolventCubicSolutions) R_m = null #R_m = resolventCubicSolutions.tensor.elem[1] for eachSolution in resolventCubicSolutions.tensor.elem if DEBUG then console.log("examining solution: " + eachSolution) push(eachSolution) push_integer(2) multiply() push(R_p) add() absValFloat() toBeCheckedIFZero = pop() if DEBUG then console.log("abs value is: " + eachSolution) if !iszero(toBeCheckedIFZero) R_m = eachSolution break if DEBUG then console.log("chosen solution: " + R_m) push(R_m) push_integer(2) multiply() push(R_p) add() push_rational(1,2) power() simplify() sqrtPPlus2M = pop() push(R_q) push_integer(2) multiply() push(sqrtPPlus2M) divide() simplify() TwoQOversqrtPPlus2M = pop() push(R_p) push_integer(3) multiply() push(R_m) push_integer(2) multiply() add() ThreePPlus2M = pop() # solution1 push(sqrtPPlus2M) push(ThreePPlus2M) push(TwoQOversqrtPPlus2M) add() negate() push_rational(1,2) power() simplify() add() push_integer(2) divide() # solution2 push(sqrtPPlus2M) push(ThreePPlus2M) push(TwoQOversqrtPPlus2M) add() negate() push_rational(1,2) power() simplify() subtract() push_integer(2) divide() # solution3 push(sqrtPPlus2M) negate() push(ThreePPlus2M) push(TwoQOversqrtPPlus2M) subtract() negate() push_rational(1,2) power() simplify() add() push_integer(2) divide() # solution4 push(sqrtPPlus2M) negate() push(ThreePPlus2M) push(TwoQOversqrtPPlus2M) subtract() negate() push_rational(1,2) power() simplify() subtract() push_integer(2) divide() restore() return # Q --------------------------- push(R_determinant) simplify() absValFloat() R_determinant_simplified_toCheckIfZero = pop() push(R_DELTA0) simplify() absValFloat() R_DELTA0_simplified_toCheckIfZero = pop() S_CHECKED_AS_NOT_ZERO = false choiceOfRadicalInQSoSIsNotZero = 0 while !S_CHECKED_AS_NOT_ZERO Q_CHECKED_AS_NOT_ZERO = false flipSignOFRadicalSoQIsNotZero = false # Q will go as denominator to calculate S # so we have to check # that is not zero while !Q_CHECKED_AS_NOT_ZERO # D1 under the outer radical push(R_DELTA1) # D1^2 under the inner radical push(R_DELTA1) push_integer(2) power() # 4*D0^3 under the inner radical push_integer(-4) push(R_DELTA0) push_integer(3) power() multiply() # addition under the inner radical add() # the second radical push_rational(1,2) power() if flipSignOFRadicalSoQIsNotZero negate() # the addition under the outer radical add() # content of outer radical divided by two push_integer(2) divide() if DEBUG then console.log("content of cubic root: " + stack[tos-1].toString()) # outer radical calculation: cubic root # now we actually have to find all the roots # because we have to pick the one that makes S != 0 push_rational(1,3) power() simplify() R_principalCubicRoot = pop() if DEBUG then console.log("principal cubic root: " + R_principalCubicRoot.toString()) if DEBUG then console.log("tos : " + tos) if choiceOfRadicalInQSoSIsNotZero == 0 if DEBUG then console.log("chosing principal cubic root") push(R_principalCubicRoot) else if choiceOfRadicalInQSoSIsNotZero == 1 if DEBUG then console.log("chosing cubic root beyond principal") push(R_principalCubicRoot) push_rational(-1,2) multiply() push_integer(3) push_rational(1,2) power() push(imaginaryunit) multiply() push_rational(-1,2) multiply() push(R_principalCubicRoot) multiply() add() else if choiceOfRadicalInQSoSIsNotZero == 1 if DEBUG then console.log("chosing cubic root beyond beyond principal") push(R_principalCubicRoot) push_rational(-1,2) multiply() push_integer(3) push_rational(1,2) power() push(imaginaryunit) multiply() push_rational(1,2) multiply() push(R_principalCubicRoot) multiply() add() simplify() R_Q = pop() if DEBUG then console.log "Q " + R_Q.toString() if DEBUG then console.log("tos: " + tos) push(R_Q) simplify() absValFloat() R_Q_simplified_toCheckIfZero = pop() if DEBUG then console.log "Q simplified and abs" + R_Q_simplified_toCheckIfZero.toString() if iszero(R_Q_simplified_toCheckIfZero) and (!iszero(R_determinant_simplified_toCheckIfZero) and iszero(R_DELTA0_simplified_toCheckIfZero)) if DEBUG then console.log " *********************************** Q IS ZERO and it matters, flipping the sign" flipSignOFRadicalSoQIsNotZero = true else Q_CHECKED_AS_NOT_ZERO = true if DEBUG then console.log("tos: " + tos) # S push_rational(-2,3) push(R_p) multiply() push(R_Q) push(R_DELTA0) push(R_Q) divide() add() #rationalize() #console.log("rationalised: " + stack[tos-1].toString()) #simplify() push(R_3_a) divide() add() push_rational(1,2) power() push_integer(2) divide() show_power_debug = true simplify() R_S = pop() if DEBUG then console.log "S " + R_S.toString() # now check if S is zero push(R_S) simplify() absValFloat() R_S_simplified_toCheckIfZero = pop() if DEBUG then console.log "S " + R_S_simplified_toCheckIfZero.toString() if iszero(R_S_simplified_toCheckIfZero) if DEBUG then console.log " *********************************** S IS ZERO chosing another cubic root" choiceOfRadicalInQSoSIsNotZero++ else S_CHECKED_AS_NOT_ZERO = true if DEBUG then console.log("tos: " + tos) # ---------------------------- if DEBUG then console.log("tos: " + tos) push(p4) negate() push(p3) push_integer(4) multiply() divide() R_minus_b_over_4a = pop() push_integer(-4) push(R_S) push_integer(2) power() multiply() push_integer(2) push(R_p) multiply() subtract() R_minus_4S2_minus_2p = pop() push(R_q) push(R_S) divide() R_q_over_S = pop() if DEBUG then console.log("tos before putting together the 4 solutions: " + tos) # first solution push(R_minus_b_over_4a) # first term push(R_S) subtract() push(R_minus_4S2_minus_2p) push(R_q_over_S) add() push_rational(1,2) power() push_integer(2) divide() add() simplify() # second solution push(R_minus_b_over_4a) # first term push(R_S) subtract() push(R_minus_4S2_minus_2p) push(R_q_over_S) add() push_rational(1,2) power() push_integer(2) divide() subtract() simplify() # third solution push(R_minus_b_over_4a) # first term push(R_S) add() push(R_minus_4S2_minus_2p) push(R_q_over_S) subtract() push_rational(1,2) power() push_integer(2) divide() add() simplify() # fourth solution push(R_minus_b_over_4a) # first term push(R_S) add() push(R_minus_4S2_minus_2p) push(R_q_over_S) subtract() push_rational(1,2) power() push_integer(2) divide() subtract() simplify() restore() return tos -= n restore()