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cubiceq.c
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//
// This file is part of Alpertron Calculators.
//
// Copyright 2019-2024 Dario Alejandro Alpern
//
// Alpertron Calculators is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// Alpertron Calculators is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with Alpertron Calculators. If not, see <http://www.gnu.org/licenses/>.
//
#include <string.h>
#include <math.h>
#include <assert.h>
#include "rootseq.h"
#include "expression.h"
extern BigInteger Cubic;
extern BigInteger Quadratic;
extern BigInteger Linear;
extern BigInteger Independent;
extern char* ptrOutput;
extern const char* ptrACos;
extern const char* ptrCos;
extern int eqNbr;
static void CbrtIndep(void)
{
if ((pretty != PARI_GP) || !BigIntIsOne(&RatDeprIndependent.denominator))
{
startCbrt();
showRationalNoParen(&RatDeprIndependent);
endCbrt();
}
else
{
showRationalNoParen(&RatDeprIndependent);
showText("^(1/3)");
}
}
static void showRAndS(void)
{
BigRationalDivideByInt(&RatDeprIndependent, -2, &Rat3);
intToBigInteger(&Rat1.numerator, 1);
intToBigInteger(&Rat1.denominator, 1);
BigRationalDivideByInt(&RatDiscr, 4, &Rat2);
MultiplyRationalBySqrtRational(&Rat1, &Rat2);
for (int ctr = 0; ctr < 2; ctr++)
{
if (teach)
{
showText("<p>");
}
else
{
startLine();
}
if (pretty == PRETTY_PRINT)
{
showText("<var>");
}
*ptrOutput = ((ctr == 0) ? 'r' : 's');
ptrOutput++;
if (pretty == PRETTY_PRINT)
{
showText("</var>");
}
*ptrOutput = ' ';
ptrOutput++;
*ptrOutput = '=';
ptrOutput++;
*ptrOutput = ' ';
ptrOutput++;
// Pari-GP requires the argument of the cube root to be positive.
if ((Rat2.numerator.nbrLimbs == 1) && (Rat2.numerator.limbs[0].x == 1) &&
(Rat2.denominator.nbrLimbs == 1) && (Rat2.denominator.limbs[0].x == 1))
{ // Cube root of rational number.
if (ctr == 0)
{
BigRationalAdd(&Rat3, &Rat1, &Rat4);
}
else
{
BigRationalSubt(&Rat3, &Rat1, &Rat4);
}
ForceDenominatorPositive(&Rat4);
if (Rat4.numerator.sign == SIGN_NEGATIVE)
{
showText(ptrMinus);
Rat4.numerator.sign = SIGN_POSITIVE;
}
startCbrt();
showRationalNoParen(&Rat4);
endCbrt();
}
else
{ // Cube root of rational plus/minus a square root.
bool isCbrtNegative;
ForceDenominatorPositive(&Rat3);
double dLogRat = logBigNbr(&Rat3.numerator) - logBigNbr(&Rat3.denominator);
double dLogSqrt =
(logBigNbr(&Rat1.numerator) - logBigNbr(&Rat1.denominator)) +
(logBigNbr(&Rat2.numerator) - logBigNbr(&Rat2.denominator)) / 2.0;
if (ctr == 0)
{ // Square root positive
isCbrtNegative = ((Rat3.numerator.sign == SIGN_NEGATIVE) &&
(dLogRat > dLogSqrt));
}
else
{
isCbrtNegative = ((Rat3.numerator.sign == SIGN_NEGATIVE) ||
(dLogRat < dLogSqrt));
}
if (isCbrtNegative)
{
showText(ptrMinus);
BigIntChSign(&Rat3.numerator);
}
startCbrt();
showRationalNoParen(&Rat3);
if (isCbrtNegative)
{
BigIntChSign(&Rat3.numerator);
}
showPlusSignOn(ctr == (isCbrtNegative ? 1 : 0),
TYPE_PM_SPACE_BEFORE | TYPE_PM_SPACE_AFTER);
ShowRationalAndSqrParts(&Rat1, &Rat2, 2, ptrTimes);
endCbrt();
}
if (teach)
{
showText("</p>");
}
else
{
endLine();
}
}
}
static void showCosT(void)
{
showText(ptrCos);
showText(" ");
showVariable(&ptrOutput, 't');
}
static void showCosTPower3(void)
{
startParen();
showCosT();
endParen();
showPower(&ptrOutput, 3);
}
static void showArgumentArcCos(void)
{
BigRationalDivide(&RatDeprIndependent, &RatDeprLinear, &Rat1); // q/p
BigRationalMultiplyByInt(&Rat1, 3, &Rat1); // 3q/p
BigRationalDivideByInt(&Rat1, 2, &Rat1); // 3q/(2p)
ForceDenominatorPositive(&Rat1);
CopyBigInt(&Rat2.numerator, &RatDeprLinear.denominator);
CopyBigInt(&Rat2.denominator, &RatDeprLinear.numerator); // 1/p
BigRationalMultiplyByInt(&Rat2, -3, &Rat2); // -3/p
ForceDenominatorPositive(&Rat2);
MultiplyRationalBySqrtRational(&Rat1, &Rat2);
ShowRationalAndSqrParts(&Rat1, &Rat2, 2, ptrTimes);
}
static void CasusIrreducibilis(int multiplicity, char currLetter, bool fromQuartic)
{
enum eSign signRat1;
if (teach)
{
char buf[1000];
char* ptrBuf;
showText(lang ? "<p>El discriminante es positivo, lo que indica que las tres raíces son reales. "
"En este caso no es posible representar las raíces mediante expresiones radicales de números reales, "
"por lo que se debe utilizar trigonometría.</p>"
"<p>Comenzando con la fórmula de la triplicación del ángulo:</p><p>4" :
"<p>The discriminant is positive, which implies that all three roots are real. "
"In this case the roots cannot be represented by radical expressions of real numbers, "
"so we must use trigonometry.</p>"
"<p>Starting with the formula of the triple angle:</p><p>4");
showText(ptrTimes);
showCosTPower3();
showText(" ");
showText(ptrMinus);
showText(" 3");
showText(ptrTimes);
showCosT();
showText(" ");
showText(ptrMinus);
showText(" ");
showText(ptrCos);
startParen();
showText("3");
showText(ptrTimes);
showVariable(&ptrOutput, 't');
endParen();
showText(" = 0");
generateEqNbr(); // Equation 2.
showText("</p><p>");
showText(lang ? "Sea " : "Let ");
showVariable(&ptrOutput, currLetter);
showText(" = ");
showVariable(&ptrOutput, 'u');
showText(" ");
showText(ptrTimes);
showCosT();
generateEqNbr(); // Equation 3.
showText("</p><p>");
showText(lang ? "Sustituyendo (": "Replacing (");
int2dec(&ptrOutput, eqNbr); // Equation 3.
showText(lang ? ") en (" : ") in (");
int2dec(&ptrOutput, eqNbr - 2); // Equation 1.
showText("):</p><p>");
showPowerVar(&ptrOutput, 3, 'u');
showText(ptrTimes);
showCosTPower3();
showRatCoeffAndPowerVar(&RatDeprLinear, 1, 'u');
showText(ptrTimes);
showText(" ");
showCosT();
showPlusMinusRational(&RatDeprIndependent);
showText(" = 0");
generateEqNbr(); // Equation 4.
showText("</p><p>");
showText(lang ? "El criterio a seguir consiste en igualar los términos de " :
"We will equate the terms of ");
showEqNbrs(eqNbr - 2, eqNbr); // Equations 2 and 4.
showText(lang ? " de izquierda a derecha. Dividiendo por " :
" from left to right. Dividing by ");
showPowerVar(&ptrOutput, 3, 'u');
showText(" / 4:</p>4");
showCosTPower3();
BigRationalMultiplyByInt(&RatDeprLinear, 4, &Rat1);
showText(ptrTimes);
ptrBuf = buf;
showPowerVar(&ptrBuf, 2, 'u');
*ptrBuf = 0;
showPlusSignOn(Rat1.numerator.sign == SIGN_POSITIVE,
TYPE_PM_SPACE_BEFORE|TYPE_PM_SPACE_AFTER);
Rat1.numerator.sign = SIGN_POSITIVE;
showRationalOverStr(&Rat1, buf, ptrTimes);
showText(ptrTimes);
showCosT();
BigRationalMultiplyByInt(&RatDeprIndependent, 4, &Rat1);
ptrBuf = buf;
showPowerVar(&ptrBuf, 3, 'u');
*ptrBuf = 0;
showPlusSignOn(Rat1.numerator.sign == SIGN_POSITIVE,
TYPE_PM_SPACE_BEFORE | TYPE_PM_SPACE_AFTER);
Rat1.numerator.sign = SIGN_POSITIVE;
showRationalOverStr(&Rat1, buf, ptrTimes);
showText(" = 0</p><p>");
showText(lang ? "El segundo coeficiente debe valer " :
"The second coefficient must equal ");
showText(ptrMinus);
showText(lang ? "3, así que:</p><p>" : "3, so:</p><p>");
showVariable(&ptrOutput, 'u');
showText(" = ");
intToBigInteger(&Rat1.numerator, 2);
intToBigInteger(&Rat1.denominator, 1); // 2
BigRationalDivideByInt(&RatDeprLinear, -3, &Rat2); // -p/3
MultiplyRationalBySqrtRational(&Rat1, &Rat2);
ShowRationalAndSqrParts(&Rat1, &Rat2, 2, ptrTimes);
generateEqNbr(); // Equation 6.
showText("</p><p>");
showText(lang ? "Igualando el último término:" : "Equating the last term:");
showText("</p><p>");
showText(ptrCos);
startParen();
showText("3");
showText(ptrTimes);
showVariable(&ptrOutput, 't');
endParen();
showText(" = ");
BigRationalMultiplyByInt(&RatDeprIndependent, -4, &Rat1);
ptrBuf = buf;
showPowerVar(&ptrBuf, 3, 'u');
*ptrBuf = 0;
if (Rat1.numerator.sign == SIGN_NEGATIVE)
{
showText(" ");
showText(ptrMinus);
}
showText(" ");
Rat1.numerator.sign = SIGN_POSITIVE;
showRationalOverStr(&Rat1, buf, ptrTimes);
showText(" = ");
showArgumentArcCos();
showText("</p>");
}
if (teach)
{
showText("<p>");
}
else
{
startLine();
}
if (pretty == PRETTY_PRINT)
{
showText("<var>t</var> = ");
}
else
{
showText("t = ");
}
if (pretty != PARI_GP)
{
showRatConstants("1", "3");
showText(ptrTimes);
}
else
{
showText("(1/3) * ");
}
showText(ptrACos);
startParen();
showArgumentArcCos();
endParen();
if (teach)
{
showText("</p>");
}
else
{
endLine();
}
intToBigInteger(&Rat1.numerator, 2);
intToBigInteger(&Rat1.denominator, 1); // 2
BigRationalDivideByInt(&RatDeprLinear, -3, &Rat2); // -p/3
MultiplyRationalBySqrtRational(&Rat1, &Rat2);
signRat1 = Rat1.numerator.sign;
Rat1.numerator.sign = SIGN_POSITIVE;
if (teach)
{
showText(lang ? "<p>De " : "<p>From ");
showEqNbrs(eqNbr - 2, eqNbr); // Equations 4 and 6.
showText(":</p>");
}
for (int ctr = 0; ctr <= 4; ctr += 2)
{
if (fromQuartic)
{
showText("<p>");
showVariable(&ptrOutput, 'm');
showText(" = ");
}
else
{
showX(multiplicity);
}
if (!BigIntIsZero(&Quadratic))
{
showRationalNoParen(&RatQuadratic);
if (signRat1 == SIGN_POSITIVE)
{
showText(" + ");
}
}
if (signRat1 == SIGN_NEGATIVE)
{
*ptrOutput = ' ';
ptrOutput++;
showText(ptrMinus);
*ptrOutput = ' ';
ptrOutput++;
}
ShowRationalAndSqrParts(&Rat1, &Rat2, 2, ptrTimes);
showText(ptrTimes);
*ptrOutput = ' ';
ptrOutput++;
showText(ptrCos);
if (ctr == 0)
{
showText("(<var>t</var>)");
}
else
{
startParen();
showText("<var>t</var> + ");
if (ctr == 2)
{
if (pretty != PARI_GP)
{
char numer[200];
char* ptrNumer = numer;
*ptrNumer = '2';
ptrNumer++;
showTimesPi(&ptrNumer);
showRatConstants(numer, "3");
}
else
{
showText("2 * Pi / 3");
}
}
if (ctr == 4)
{
if (pretty != PARI_GP)
{
char numer[200];
char* ptrNumer = numer;
*ptrNumer = '4';
ptrNumer++;
showTimesPi(&ptrNumer);
showRatConstants(numer, "3");
}
else
{
showText("4 * Pi / 3");
}
}
endParen();
if (pretty == TEX)
{
*ptrOutput = '}';
ptrOutput++;
}
}
if (fromQuartic)
{
return;
}
endShowX();
}
}
static void showImagPartCubicRoot1(void)
{
if (pretty != PARI_GP)
{ // Show i/2.
showRatConstants("i", "2");
}
else
{
showText("I/2");
}
showText(ptrTimes);
if (pretty != PARI_GP)
{ // square root of 3.
startSqrt();
*ptrOutput = '3';
ptrOutput++;
endSqrt();
}
else
{ // square root of 3.
showText("3^(1/2)");
}
}
static void showCompleteCbrt1(const char* pszPlusMinus)
{
showText(ptrMinus);
showText(" ");
showRatConstants("1", "2");
showText(" ");
showText(pszPlusMinus);
showText(" ");
showImagPartCubicRoot1();
}
static void showNonRealCubeRootsOf1(void)
{
for (char letter = 'e'; letter <= 'f'; letter++)
{
if (letter == 'f')
{
showText(", ");
}
showVariable(&ptrOutput, letter);
showText(" = ");
showCompleteCbrt1((letter == 'e') ? "+" : ptrMinus);
}
}
// The first root x1 is q ^ (1/3)
// The second root x2 is (-1/2) * x1 + i * (3 ^ (1/2) * x1) / 2
// The third root x3 is (-1/2) * x1 - i * (3 ^ (1/2) * x1) / 2
static void bothQuadraticAndLinearCoeffZero(int multiplicity, bool fromQuartic)
{
bool isCbrtNegative;
BigRationalNegate(&RatDeprIndependent, &RatDeprIndependent);
ForceDenominatorPositive(&RatDeprIndependent);
if (teach)
{
showText(lang ? "<p>Las soluciones son la raíz cúbica real de " :
"<p>The solutions are the real cube root of ");
showRationalNoParen(&RatDeprIndependent);
showText(lang ? " y su producto por las dos raíces cúbicas no reales de 1, que son:</p><p>":
" and the multiplication by both non-real cube roots of 1:</p><p>");
showNonRealCubeRootsOf1();
showText("</p>");
}
isCbrtNegative = (RatDeprIndependent.numerator.sign == SIGN_NEGATIVE);
RatDeprIndependent.numerator.sign = SIGN_POSITIVE;
for (int ctr = 0; ctr < 3; ctr++)
{
showX(multiplicity);
if (BigIntIsZero(&Quadratic))
{
if ((ctr == 0) && isCbrtNegative)
{
showText(ptrMinus);
}
}
else
{
showRationalNoParen(&RatQuadratic);
if ((ctr == 0) && isCbrtNegative)
{
*ptrOutput = ' ';
ptrOutput++;
showText(ptrMinus);
*ptrOutput = ' ';
ptrOutput++;
}
else
{
showText(" + ");
}
}
if (ctr == 0)
{
CbrtIndep(); // q^(1/3)
}
else
{
if (pretty != PARI_GP)
{
if (!isCbrtNegative)
{
showText(ptrMinus);
}
*ptrOutput = ' ';
ptrOutput++;
showRatConstants("1", "2");
*ptrOutput = ' ';
ptrOutput++;
showText(ptrTimes);
}
else
{
if (isCbrtNegative)
{
showText("(1/2) *");
}
else
{
showText("(-1/2) *");
}
}
CbrtIndep(); // q^(1/3)
showPlusSignOn(ctr == 1, TYPE_PM_SPACE_BEFORE | TYPE_PM_SPACE_AFTER);
showImagPartCubicRoot1();
*ptrOutput = ' ';
ptrOutput++;
showText(ptrTimes);
CbrtIndep(); // q^(1/3)
}
endShowX();
}
}
static void showRPlusS(void)
{
startParen();
showVariable(&ptrOutput, 'r');
showText(" + ");
showVariable(&ptrOutput, 's');
endParen();
}
static void showN(void)
{
showCoeffBeforeParen(&RatDeprLinear);
showRPlusS();
showPlusMinusRational(&RatDeprIndependent);
showText(" = 0</p><p>");
}
static void showR(void)
{
showVariable(&ptrOutput, 'r');
}
static void showCardanoMethod(char currLetter)
{
showText(lang ? "<p>Usando el método de Cardano, asignando " : "<p>Using Cardano's method, setting ");
showVariable(&ptrOutput, currLetter);
showText(" = ");
showVariable(&ptrOutput, 'r');
showText(" + ");
showVariable(&ptrOutput, 's');
showText(":</p><p>");
// Show (r+s)^3 + N = 0, where N = p(r+s) + q
showRPlusS();
showPower(&ptrOutput, 3);
showN();
// Show r^3 + 3r^2*s + 3rs^2 + s^3 + N = 0
showPowerVar(&ptrOutput, 3, 'r');
intToBigInteger(&Rat1.numerator, 3);
intToBigInteger(&Rat1.denominator, 1);
showRatCoeffAndPowerVar(&Rat1, 2, 'r');
showText(ptrTimes);
showVariable(&ptrOutput, 's');
showRatCoeffAndPowerVar(&Rat1, 1, 'r');
showText(ptrTimes);
showPowerVar(&ptrOutput, 2, 's');
showText(" + ");
showPowerVar(&ptrOutput, 3, 's');
showN();
// Show r^3 + s^3 + 3rs(r+s) + N = 0
showPowerVar(&ptrOutput, 3, 'r');
showText(" + ");
showPowerVar(&ptrOutput, 3, 's');
showRatCoeffAndPowerVar(&Rat1, 1, 'r');
showText(ptrTimes);
showVariable(&ptrOutput, 's');
showText(ptrTimes);
showRPlusS();
showN();
// Show r^3 + s^3 + (3rs + p)(r+s) + q = 0
showPowerVar(&ptrOutput, 3, 'r');
showText(" + ");
showPowerVar(&ptrOutput, 3, 's');
showText(" + ");
startParen();
showRational(&Rat1);
showText(ptrTimes);
showVariable(&ptrOutput, 'r');
showText(ptrTimes);
showVariable(&ptrOutput, 's');
showPlusMinusRational(&RatDeprLinear);
endParen();
showText(ptrTimes);
showRPlusS();
showPlusMinusRational(&RatDeprIndependent);
showText(" = 0");
generateEqNbr(); // Equation 1
showText("</p><p>");
showText(lang ? "Como hay una variable adicional, se puede imponer una condición extra. En este caso la elección es:</p><p>" :
"Since there is an extra variable, we can impose an additional condition. In our case it is:</p><p>");
// Show 3rs + p = 0
showRational(&Rat1);
showText(ptrTimes);
showVariable(&ptrOutput, 'r');
showText(ptrTimes);
showVariable(&ptrOutput, 's');
showPlusMinusRational(&RatDeprLinear);
showText(" = 0");
generateEqNbr(); // Equation 2
showText("</p><p>");
// Show rs = -p/3
showVariable(&ptrOutput, 'r');
showText(ptrTimes);
showVariable(&ptrOutput, 's');
showText(" = ");
BigRationalDivideByInt(&RatDeprLinear, -3, &Rat1);
ForceDenominatorPositive(&Rat1);
showRational(&Rat1);
generateEqNbr(); // Equation 3
showText("</p><p>");
showPowerVar(&ptrOutput, 3, 'r');
showText(ptrTimes);
showPowerVar(&ptrOutput, 3, 's');
showText(" = ");
BigRationalMultiply(&Rat1, &Rat1, &Rat2);
BigRationalMultiply(&Rat2, &Rat1, &Rat2);
showRational(&Rat2);
generateEqNbr(); // Equation 4
showText("</p><p>");
showText(lang ? "De " : "From ");
showEqNbrs(eqNbr - 3, eqNbr - 2); // Equations 1 and 2.
showText(":</p><p>");
// Show r^3 + s^3 + q = 0
showPowerVar(&ptrOutput, 3, 'r');
showText(" + ");
showPowerVar(&ptrOutput, 3, 's');
showPlusMinusRational(&RatDeprIndependent);
showText(" = 0");
generateEqNbr(); // Equation 5
showText("</p><p>");
showText(lang ? "Multiplicando por " : "Multiplying by ");
showPowerVar(&ptrOutput, 3, 'r');
showText(":</p></p>");
// Show r^6 + r^3*s^3 + q*r^3 = 0
showPowerVar(&ptrOutput, 6, 'r');
showText(" + ");
showPowerVar(&ptrOutput, 3, 'r');
showText(ptrTimes);
showPowerVar(&ptrOutput, 3, 's');
showRatCoeffAndPowerVar(&RatDeprIndependent, 3, 'r');
showText(" = 0</p><p>");
showText(lang ? "De (": "From (");
int2dec(&ptrOutput, eqNbr - 1); // Equation 4.
showText("):</p>");
showPowerVar(&ptrOutput, 6, 'r');
showPlusMinusRational(&Rat2);
showRatCoeffAndPowerVar(&RatDeprIndependent, 3, 'r');
showText(" = 0</p><p>");
showPowerVar(&ptrOutput, 6, 'r');
showRatCoeffAndPowerVar(&RatDeprIndependent, 3, 'r');
showPlusMinusRational(&Rat2);
showText(" = 0</p><p>");
showText(lang ? "Esta es una ecuación cuadrática en " :
"This is a quadratic equation in ");
showPowerVar(&ptrOutput, 3, 'r');
showText(lang ? ". Al multiplicar (" : ". If we multiplied (");
int2dec(&ptrOutput, eqNbr); // Equation 5.
showText(lang ? ") por " : ") by ");
showPowerVar(&ptrOutput, 3, 's');
showText(lang ? " en vez de " : " instead of ");
showPowerVar(&ptrOutput, 3, 'r');
showText(lang ? ", los coeficientes de la ecuación serían los mismos, así que la ecuación cuadrática también permite obtener " :
", the equation coefficients would be the same, so the quadratic equation can also give us the value of ");
showPowerVar(&ptrOutput, 3, 's');
showText(lang ? ". Sea " : ". Let ");
showVariable(&ptrOutput, 'w');
showText(" = ");
showPowerVar(&ptrOutput, 3, 'r');
showText(lang ? " o " : " or ");
showPowerVar(&ptrOutput, 3, 's');
showText(".</p><p>");
// Back-up linear and independent coefficients.
CopyBigInt(&RatQuartic.numerator, &RatLinear.numerator);
CopyBigInt(&RatQuartic.denominator, &RatLinear.denominator);
CopyBigInt(&RatCubic.numerator, &RatIndependent.numerator);
CopyBigInt(&RatCubic.denominator, &RatIndependent.denominator);
// Populate linear and independent coefficients.
CopyBigInt(&RatLinear.numerator, &RatDeprIndependent.numerator);
CopyBigInt(&RatLinear.denominator, &RatDeprIndependent.denominator);
CopyBigInt(&RatIndependent.numerator, &Rat2.numerator);
CopyBigInt(&RatIndependent.denominator, &Rat2.denominator);
showRatCoeffAndPowerVar(NULL, -2, 'w');
showRatCoeffAndPowerVar(&RatLinear, 1, 'w');
showRatCoeffAndPowerVar(&RatIndependent, 0, 'w');
showText(" = 0</p>");
stepsForQuadraticEquation('w', 'z');
// Restore linear and independent coefficients.
CopyBigInt(&RatLinear.numerator, &RatQuartic.numerator);
CopyBigInt(&RatLinear.denominator, &RatQuartic.denominator);
CopyBigInt(&RatIndependent.numerator, &RatCubic.numerator);
CopyBigInt(&RatIndependent.denominator, &RatCubic.denominator);
}
static void linearCoeffNotZero(int multiplicity, char currLetter, bool fromQuartic)
{
if (teach)
{
showText(lang ? "<p>El discriminante es negativo, lo que indica que hay una raíz real y dos raíces complejas conjugadas.</p>" :
"<p>The discriminant is negative, so there is a real root and two complex conjugate roots.</p>");
showCardanoMethod(currLetter);
}
showRAndS();
if (!fromQuartic && teach)
{
showText(lang ? "<p>Una raíz cúbica tiene tres soluciones en el campo complejo. Sin embargo, no se puede elegir cualquier valor para " :
"<p>A cubic root has three solutions in the complex field. But we cannot select any value for ");
showVariable(&ptrOutput, 'r');
showText(lang ? " y " : " and ");
showVariable(&ptrOutput, 's');
showText(lang ? " porque se debe cumplir la condición (" : "because the condition (");
int2dec(&ptrOutput, eqNbr - 2); // Equation 3.
showText(lang ? "). Así que el producto debe ser un número real.</p><p>Sea " :
") must be true. So their product must be a real number.</p><p>Let ");
for (char letter = 'r'; letter <= 's'; letter++)
{
if (letter == 's')
{
showText(", ");
}
showVarIndex(letter, 1);
showText(" = ");
showVariable(&ptrOutput, letter);
showText(", ");
showVarIndex(letter, 2);
showText(" = ");
showVariable(&ptrOutput, letter);
showText(ptrTimes);
showVariable(&ptrOutput, 'e');
showText(", ");
showVarIndex(letter, 3);
showText(" = ");
showVariable(&ptrOutput, letter);
showText(ptrTimes);
showVariable(&ptrOutput, 'f');
}
showText(lang ? ", donde</p><p>" : ", where</p><p>");
showNonRealCubeRootsOf1();
showText(lang ? "</p><p>son las raíces cúbicas no reales de 1. Como " :
"</p><p>are the non-real cubic roots of 1. Since ");
showVariable(&ptrOutput, 'e');
showText(ptrTimes);
showVariable(&ptrOutput, 'f');
showText(lang ? " es real, pero " : " is real, but ");
showPowerVar(&ptrOutput, 2, 'e');
showText(lang ? " y " : " and ");
showPowerVar(&ptrOutput, 2, 'f');
showText(lang ? " no lo son, los valores de " : " are not, the values of ");
showVariable(&ptrOutput, currLetter);
showText(lang ? " que cumplen la condición (" : " that follow the condition (");
int2dec(&ptrOutput, eqNbr - 2); // Equation 3.
showText(lang ? ") son:</p>" : ") are:</p>");
for (int rootNbr = 1; rootNbr <= 3; rootNbr++)
{
showText("<p>");
if (currLetter == 'y')
{
showVarIndex('y', rootNbr);
showText(" = ");
}
showVarIndex('x', indexRoot + (rootNbr - 1) * multiplicity);
if (!BigIntIsZero(&RatQuadratic.numerator))
{
BigIntChSign(&RatQuadratic.numerator);
showPlusMinusRational(&RatQuadratic);
BigIntChSign(&RatQuadratic.numerator);
}
showText(" = ");
switch (rootNbr)
{
case 1:
showVarIndex('r', 1);
showText(" + ");
showVarIndex('s', 1);
showText(" = ");
showVariable(&ptrOutput, 'r');
showText(" + ");
showVariable(&ptrOutput, 's');
break;
default:
showVarIndex('r', rootNbr);
showText(" + ");
showVarIndex('s', 5 - rootNbr);
showText(" = ");
for (char letter = 'r'; letter <= 's'; letter++)
{
showVariable(&ptrOutput, letter);
showText(ptrTimes);
*ptrOutput = ' ';
ptrOutput++;
startParen();
if ((rootNbr == 2) == (letter == 'r'))
{
showCompleteCbrt1("+");
}
else
{
showCompleteCbrt1(ptrMinus);
}
endParen();
if (letter == 'r')
{
showText(" + ");
}
}
break;
}
showText("</p>");
}
}
for (int ctr = 0; ctr < 3; ctr++)
{
if (fromQuartic)
{
showText("<p>");
showVariable(&ptrOutput, 'm');
showText(" = ");
}
else
{
showX(multiplicity);
}
if (!BigIntIsZero(&Quadratic))
{
showRationalNoParen(&RatQuadratic);
showPlusSignOn(ctr == 0, TYPE_PM_SPACE_BEFORE | TYPE_PM_SPACE_AFTER);
}
if (ctr == 0)
{
showText("<var>r</var> + <var>s</var>");
}
else
{
if (BigIntIsZero(&RatQuadratic.numerator))
{
showText(ptrMinus);
}
if (pretty == PRETTY_PRINT)
{
showRatConstants("<var>r</var> + <var>s</var>", "2");
}
else if (pretty == TEX)
{
showRatConstants("r + s", "2");
}
else
{
showText("(<var>r</var> + <var>s</var>) / 2");
}
showPlusSignOn(ctr == 1, TYPE_PM_SPACE_BEFORE | TYPE_PM_SPACE_AFTER);
if (pretty != PARI_GP)
{
showText("i");
*ptrOutput = ' ';
ptrOutput++;
showText(ptrTimes);
*ptrOutput = ' ';
ptrOutput++;
showRatConstants((pretty == TEX) ? "r - s" :
"<var>r</var> − <var>s</var>", "2");
showText(ptrTimes);
startSqrt();
*ptrOutput = '3';
ptrOutput++;
endSqrt();
}
else
{
showText("(I/2) * (<var>r</var> - <var>s</var>) * 3^(1/2)");
}
}
if (fromQuartic)
{
showText(lang? "<p>De (": "<p>From (");
int2dec(&ptrOutput, eqNbr - 2);
showText("):</p><p>");
showVariable(&ptrOutput, 'm');
showText(" = ");
if (!BigIntIsZero(&Quadratic))
{
showRationalNoParen(&RatQuadratic);
showText(" + ");
}
showVariable(&ptrOutput, 'r');
showPlusSignOn(RatDeprLinear.numerator.sign == SIGN_NEGATIVE,
TYPE_PM_SPACE_BEFORE | TYPE_PM_SPACE_AFTER);
BigRationalDivideByInt(&RatDeprLinear, 3, &Rat1);
Rat1.numerator.sign = SIGN_POSITIVE;
showRationalNoParenOverGeneric(&Rat1, showR);
return;
}
endShowX();
}
}
void showCoeffBeforeParen(BigRational* rat)
{
if (BigIntEqual(&rat->numerator, &rat->denominator))
{
showText(" + ");
}
else
{
bool ratIsMinusOne;
BigIntChSign(&rat->numerator);
ratIsMinusOne = BigIntEqual(&rat->numerator, &rat->denominator);
BigIntChSign(&rat->numerator);
if (ratIsMinusOne)
{
showText(" ");
showText(ptrMinus);
showText(" ");
}
else
{
showPlusMinusRational(rat);
showText(ptrTimes);
}
}
}
// When coming from quartic equation, only one root is required.
void solveCubic(int multiplicity, bool fromQuartic, char currLetter)
{
// Compute coefficients of depressed equation x^3 + px + q
// where: p = -b^2/3 + c, q = 2b^3/27 - bc/3 + d.
BigRationalDivideByInt(&RatQuadratic, 3, &Rat1);
BigRationalMultiply(&Rat1, &RatQuadratic, &RatDeprLinear); // b^2/3
BigRationalMultiply(&Rat1, &RatLinear, &RatDeprIndependent); // bc/3
BigRationalMultiply(&RatDeprLinear, &RatQuadratic, &Rat1); // b^3/3
BigRationalMultiplyByInt(&Rat1, 2, &Rat1); // 2b^3/3
BigRationalDivideByInt(&Rat1, 9, &Rat1); // 2b^3/27
BigRationalSubt(&RatLinear, &RatDeprLinear, &RatDeprLinear); // p
BigRationalSubt(&RatIndependent, &RatDeprIndependent, &RatDeprIndependent); // d - bc/3
BigRationalAdd(&RatDeprIndependent, &Rat1, &RatDeprIndependent); // q
// Compute discriminant (delta)/(-27) = q^2 + 4p^3/27.