Laboratoire de l'Informatique et du Parallélisme » Pobyso

r"""

Sage core functions needed for the implementation of SLZ.

AUTHORS:

- S.T. (2013-08): initial version

Examples:

TODO::

"""

print "sageSLZ loading..."

#

def slz_check_htr_value(function, htrValue, lowerBound, upperBound, precision, \

                        degree, targetHardnessToRound, alpha):

    """

    Check an Hard-to-round value.

    """

    polyApproxPrec = targetHardnessToRound + 1

    polyTargetHardnessToRound = targetHardnessToRound + 1

    internalSollyaPrec = ceil((RR('1.5') * targetHardnessToRound) / 64) * 64

    RRR = htrValue.parent()

    #

    ## Compute the scaled function.

    fff = slz_compute_scaled_function(f, lowerBound, upperBound, precision)[0]

    print "Scaled function:", fff

    #

    ## Compute the scaling.

    boundsIntervalRifSa = RealIntervalField(precision)

    domainBoundsInterval = boundsIntervalRifSa(lowerBound, upperBound)

    scalingExpressions = \

        slz_interval_scaling_expression(domainBoundsInterval, i)

    #

    ## Get the polynomials, bounds, etc. for all the interval.

    resultListOfTuplesOfSo = \

        slz_get_intervals_and_polynomials(f, degree, lowerBound, upperBound, \

                                          precision, internalSollyaPrec,\

                                          2^-(polyApproxPrec))

    #

    ## We only want one interval.

    if len(resultListOfTuplesOfSo) > 1:

        print "Too many intervals! Aborting!"

        exit

    #

    ## Get the first tuple of Sollya objects as Sage objects.

    firstTupleSa = \

        slz_interval_and_polynomial_to_sage(resultListOfTuplesOfSo[0])

    pobyso_set_canonical_on()

    #

    print "Floatting point polynomial:", firstTupleSa[0]

    print "with coefficients precision:", firstTupleSa[0].base_ring().prec()

    #

    ## From a polynomial over a real ring, create a polynomial over the

    #  rationals ring.

    rationalPolynomial = \

        slz_float_poly_of_float_to_rat_poly_of_rat(firstTupleSa[0])

    print "Rational polynomial:", rationalPolynomial

    #

    ## Create a polynomial over the rationals that will take integer

    # variables instead of rational.

    rationalPolynomialOfIntegers = \

        slz_rat_poly_of_rat_to_rat_poly_of_int(rationalPolynomial, precision)

    print "Type:", type(rationalPolynomialOfIntegers)

    print "Rational polynomial of integers:", rationalPolynomialOfIntegers

    #

    ## Check the rational polynomial of integers variables.

    # (check against the scaled function).

    toIntegerFactor = 2^(precision-1)

    intervalCenterAsIntegerSa = int(firstTupleSa[3] * toIntegerFactor)

    print "Interval center as integer:", intervalCenterAsIntegerSa

    lowerBoundAsIntegerSa = int(firstTupleSa[2].endpoints()[0] * \

                                toIntegerFactor) - intervalCenterAsIntegerSa

    upperBoundAsIntegerSa = int(firstTupleSa[2].endpoints()[1] * \

                                toIntegerFactor) - intervalCenterAsIntegerSa

    print "Lower bound as integer:", lowerBoundAsIntegerSa

    print "Upper bound as integer:", upperBoundAsIntegerSa

    print "Image of the lower bound by the scaled function", \

            fff(firstTupleSa[2].endpoints()[0])

    print "Image of the lower bound by the approximation polynomial of ints:", \

            RRR(rationalPolynomialOfIntegers(lowerBoundAsIntegerSa))

    print "Image of the center by the scaled function", fff(firstTupleSa[3])

    print "Image of the center by the approximation polynomial of ints:", \

            RRR(rationalPolynomialOfIntegers(0))

    print "Image of the upper bound by the scaled function", \

            fff(firstTupleSa[2].endpoints()[1])

    print "Image of the upper bound by the approximation polynomial of ints:", \

            RRR(rationalPolynomialOfIntegers(upperBoundAsIntegerSa))

# End slz_check_htr_value.

#

def slz_compute_binade_bounds(number, emin):

    """

    For given "real number", compute the bounds of the binade it belongs to.

    TODO::

        Deal with the emax exponent.

    """

    RF = number.parent()

    precision = RF.precision()

    RRF = RealField(2 * precision)

    if number == 0:

        return (RF(0),RF(2^(emin)) - RF(2^(emin-precision)))

    l2 = RRF(number).abs().log2()

    offset = int(l2)

    if l2 >= 0:

        if number >= 0:

            lb = RF(2^offset)

            ub = RF(2^(offset + 1) - 2^(-precision+offset+1))

        else: #number < 0

            lb = -RF(2^(offset + 1) - 2^(-precision+offset+1))

            ub = -RF(2^offset)

    else: # log2 < 0

        if l2 < emin: # Denormal

            print "Denormal:", l2

            if number >= 0:

                lb = RF(0)

                ub = RF(2^(emin)) - RF(2^(emin-precision))

            else: # number <= 0

                lb = - RF(2^(emin)) + RF(2^(emin-precision))

                ub = RF(0)

        elif l2 > emin: # Normal number other than +/-2^emin.

            if number >= 0:

                lb = RF(2^(offset-1))

                ub = RF(2^(offset)) - RF(2^(-precision+offset))

            else: # number < 0

                lb = -RF(2^(offset) - 2^(-precision+offset))

                ub = -RF(2^(offset-1))

        else: # +/-2^emin

            if number >= 0:

                lb = RF(2^(offset))

                ub = RF(2^(offset+1)) - RF(2^(-precision+offset+1))

            else: # number < 0

                lb = -RF(2^(offset+1) - 2^(-precision+offset+1))

                ub = -RF(2^(offset))

    return (lb, ub)

# End slz_compute_binade_bounds

#

def slz_compute_polynomial_and_interval(functionSo, degreeSo, lowerBoundSa,

                                        upperBoundSa, approxPrecSa,

                                        sollyaPrecSa=None):

    """

    Under the assumptions listed for slz_get_intervals_and_polynomials, compute

    a polynomial that approximates the function on a an interval starting

    at lowerBoundSa and finishing at a value that guarantees that the polynomial

    approximates with the expected precision.

    The interval upper bound is lowered until the expected approximation

    precision is reached.

    The polynomial, the bounds, the center of the interval and the error

    are returned.

    """

    RRR = lowerBoundSa.parent()

    intervalShrinkConstFactorSa = RRR('0.5')

    absoluteErrorTypeSo = pobyso_absolute_so_so()

    currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa)

    currentUpperBoundSa = upperBoundSa

    currentLowerBoundSa = lowerBoundSa

    # What we want here is the polynomial without the variable change,

    # since our actual variable will be x-intervalCenter defined over the

    # domain [lowerBound-intervalCenter , upperBound-intervalCenter].

    (polySo, intervalCenterSo, maxErrorSo) = \

        pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo,

                                                    currentRangeSo,

                                                    absoluteErrorTypeSo)

    maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    while maxErrorSa > approxPrecSa:

        #print "++Approximation error:", maxErrorSa

        sollya_lib_clear_obj(maxErrorSo)

        sollya_lib_clear_obj(polySo)

        sollya_lib_clear_obj(intervalCenterSo)

        shrinkFactorSa = RRR('5')/(maxErrorSa/approxPrecSa).log2().abs()

        #shrinkFactorSa = 1.5/(maxErrorSa/approxPrecSa)

        #errorRatioSa = approxPrecSa/maxErrorSa

        #print "Error ratio: ", errorRatioSa

        if shrinkFactorSa > intervalShrinkConstFactorSa:

            actualShrinkFactorSa = intervalShrinkConstFactorSa

            #print "Fixed"

        else:

            actualShrinkFactorSa = shrinkFactorSa

            #print "Computed",shrinkFactorSa,maxErrorSa

            #print shrinkFactorSa, maxErrorSa

        #print "Shrink factor", actualShrinkFactorSa

        currentUpperBoundSa = currentLowerBoundSa + \

                                  (currentUpperBoundSa - currentLowerBoundSa) * \

                                   actualShrinkFactorSa

        #print "Current upper bound:", currentUpperBoundSa

        sollya_lib_clear_obj(currentRangeSo)

        sollya_lib_clear_obj(polySo)

        if currentUpperBoundSa <= currentLowerBoundSa or \

              currentUpperBoundSa == currentLowerBoundSa.nextabove():

            sollya_lib_clear_obj(absoluteErrorTypeSo)

            print "Can't find an interval."

            print "Use either or both a higher polynomial degree or a higher",

            print "internal precision."

            print "Aborting!"

            return (None, None, None, None)

        currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa,

                                                      currentUpperBoundSa)

        # print "New interval:",

        # pobyso_autoprint(currentRangeSo)

        (polySo, intervalCenterSo, maxErrorSo) = \

            pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo,

                                                        currentRangeSo,

                                                        absoluteErrorTypeSo)

        #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR)

        #print "Max errorSo:",

        #pobyso_autoprint(maxErrorSo)

        maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

        #print "Max errorSa:", maxErrorSa

        #print "Sollya prec:",

        #pobyso_autoprint(sollya_lib_get_prec(None))

    sollya_lib_clear_obj(absoluteErrorTypeSo)

    return((polySo, currentRangeSo, intervalCenterSo, maxErrorSo))

# End slz_compute_polynomial_and_interval

def slz_compute_reduced_polynomials(reducedMatrix,

                                    knownMonomials,

                                    var1,

                                    var1Bound,

                                    var2,

                                    var2Bound):

    """

    From a reduced matrix, holding the coefficients, from a monomials list,

    from the bounds of each variable, compute the corresponding polynomials

    scaled back by dividing by the "right" powers of the variables bounds.

    The elements in knownMonomials must be of the "right" polynomial type.

    They set the polynomial type of the output polynomials list.

    """

    # TODO: check input arguments.

    reducedPolynomials = []

    #print "type var1:", type(var1), " - type var2:", type(var2)

    for matrixRow in reducedMatrix.rows():

        currentPolynomial = 0

        for colIndex in xrange(0, len(knownMonomials)):

            currentCoefficient = matrixRow[colIndex]

            if currentCoefficient != 0:

                #print "Current coefficient:", currentCoefficient

                currentMonomial = knownMonomials[colIndex]

                parentRing = currentMonomial.parent()

                #print "Monomial as multivariate polynomial:", \

                #currentMonomial, type(currentMonomial)

                degreeInVar1 = currentMonomial.degree(parentRing(var1))

                #print "Degree in var", var1, ":", degreeInVar1

                degreeInVar2 = currentMonomial.degree(parentRing(var2))

                #print "Degree in var", var2, ":", degreeInVar2

                if degreeInVar1 > 0:

                    currentCoefficient = \

                        currentCoefficient / var1Bound^degreeInVar1

                    #print "varBound1 in degree:", var1Bound^degreeInVar1

                    #print "Current coefficient(1)", currentCoefficient

                if degreeInVar2 > 0:

                    currentCoefficient = \

                        currentCoefficient / var2Bound^degreeInVar2

                    #print "Current coefficient(2)", currentCoefficient

                #print "Current reduced monomial:", (currentCoefficient * \

                #                                    currentMonomial)

                currentPolynomial += (currentCoefficient * currentMonomial)

                #print "Current polynomial:", currentPolynomial

            # End if

        # End for colIndex.

        #print "Type of the current polynomial:", type(currentPolynomial)

        reducedPolynomials.append(currentPolynomial)

    return reducedPolynomials

# End slz_compute_reduced_polynomials.

def slz_compute_scaled_function(functionSa,

                                lowerBoundSa,

                                upperBoundSa,

                                floatingPointPrecSa):

    """

    From a function, compute the scaled function whose domain

    is included in [1, 2) and whose image is also included in [1,2).

    Return a tuple:

    [0]: the scaled function

    [1]: the scaled domain lower bound

    [2]: the scaled domain upper bound

    [3]: the scaled image lower bound

    [4]: the scaled image upper bound

    """

    x = functionSa.variables()[0]

    # Reassert f as a function (an not a mere expression).

    # Scalling the domain -> [1,2[.

    boundsIntervalRifSa = RealIntervalField(floatingPointPrecSa)

    domainBoundsIntervalSa = boundsIntervalRifSa(lowerBoundSa, upperBoundSa)

    (domainScalingExpressionSa, invDomainScalingExpressionSa) = \

        slz_interval_scaling_expression(domainBoundsIntervalSa, x)

    print "domainScalingExpression for argument :", domainScalingExpressionSa

    print "f: ", f

    ff = f.subs({x : domainScalingExpressionSa})

    #ff = f.subs_expr(x==domainScalingExpressionSa)

    domainScalingFunction(x) = invDomainScalingExpressionSa

    scaledLowerBoundSa = domainScalingFunction(lowerBoundSa).n()

    scaledUpperBoundSa = domainScalingFunction(upperBoundSa).n()

    print 'ff:', ff, "- Domain:", scaledLowerBoundSa, scaledUpperBoundSa

    #

    # Scalling the image -> [1,2[.

    flbSa = f(lowerBoundSa).n()

    fubSa = f(upperBoundSa).n()

    if flbSa <= fubSa: # Increasing

        imageBinadeBottomSa = floor(flbSa.log2())

    else: # Decreasing

        imageBinadeBottomSa = floor(fubSa.log2())

    print 'ff:', ff, '- Image:', flbSa, fubSa, imageBinadeBottomSa

    imageBoundsIntervalSa = boundsIntervalRifSa(flbSa, fubSa)

    (imageScalingExpressionSa, invImageScalingExpressionSa) = \

        slz_interval_scaling_expression(imageBoundsIntervalSa, x)

    iis = invImageScalingExpressionSa.function(x)

    fff = iis.subs({x:ff})

    print "fff:", fff,

    print " - Image:", fff(scaledLowerBoundSa), fff(scaledUpperBoundSa)

    return([fff, scaledLowerBoundSa, scaledUpperBoundSa, \

            fff(scaledLowerBoundSa), fff(scaledUpperBoundSa)])

def slz_float_poly_of_float_to_rat_poly_of_rat(polyOfFloat):

    # Create a polynomial over the rationals.

    polynomialRing = QQ[str(polyOfFloat.variables()[0])]

    return(polynomialRing(polyOfFloat))

# End slz_float_poly_of_float_to_rat_poly_of_rat.

def slz_get_intervals_and_polynomials(functionSa, degreeSa,

                                      lowerBoundSa,

                                      upperBoundSa, floatingPointPrecSa,

                                      internalSollyaPrecSa, approxPrecSa):

    """

    Under the assumption that:

    - functionSa is monotonic on the [lowerBoundSa, upperBoundSa] interval;

    - lowerBound and upperBound belong to the same binade.

    from a:

    - function;

    - a degree

    - a pair of bounds;

    - the floating-point precision we work on;

    - the internal Sollya precision;

    - the requested approximation error

    The initial interval is, possibly, splitted into smaller intervals.

    It return a list of tuples, each made of:

    - a first polynomial (without the changed variable f(x) = p(x-x0));

    - a second polynomial (with a changed variable f(x) = q(x))

    - the approximation interval;

    - the center, x0, of the interval;

    - the corresponding approximation error.

    TODO: fix endless looping for some parameters sets.

    """

    # Set Sollya to the necessary internal precision.

    currentSollyaPrecSo = pobyso_get_prec_so()

    currentSollyaPrecSa = pobyso_constant_from_int_so_sa(currentSollyaPrecSo)

    if internalSollyaPrecSa > currentSollyaPrecSa:

        pobyso_set_prec_sa_so(internalSollyaPrecSa)

    #

    x = functionSa.variables()[0] # Actual variable name can be anything.

    # Scaled function: [1=,2] -> [1,2].

    (fff, scaledLowerBoundSa, scaledUpperBoundSa,                             \

            scaledLowerBoundImageSa, scaledUpperBoundImageSa) =               \

                slz_compute_scaled_function(functionSa,                       \

                                            lowerBoundSa,                     \

                                            upperBoundSa,                     \

                                            floatingPointPrecSa)

    #

    resultArray = []

    #

    print "Approximation precision: ", RR(approxPrecSa)

    # Prepare the arguments for the Taylor expansion computation with Sollya.

    functionSo = pobyso_parse_string_sa_so(fff._assume_str())

    degreeSo = pobyso_constant_from_int_sa_so(degreeSa)

    scaledBoundsSo = pobyso_bounds_to_range_sa_so(scaledLowerBoundSa,

                                                  scaledUpperBoundSa)

    # Compute the first Taylor expansion.

    (polySo, boundsSo, intervalCenterSo, maxErrorSo) = \

        slz_compute_polynomial_and_interval(functionSo, degreeSo,

                                        scaledLowerBoundSa, scaledUpperBoundSa,

                                        approxPrecSa, internalSollyaPrecSa)

    if polySo is None:

        print "slz_get_intervals_and_polynomials: Aborting and returning None!"

        if internalSollyaPrecSa != currentSollyaPrecSa:

            pobyso_set_prec_sa_so(currentSollyaPrecSa)

            sollya_lib_clear_obj(currentSollyaPrecSo)

        sollya_lib_clear_obj(functionSo)

        sollya_lib_clear_obj(degreeSo)

        sollya_lib_clear_obj(scaledBoundsSo)

        return None

    realIntervalField = RealIntervalField(max(lowerBoundSa.parent().precision(),

                                              upperBoundSa.parent().precision()))

    boundsSa = pobyso_range_to_interval_so_sa(boundsSo, realIntervalField)

    errorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    #print "First approximation error:", errorSa

    # If the error and interval are OK a the first try, just return.

    if boundsSa.endpoints()[1] >= scaledUpperBoundSa:

        # Change variable stuff in Sollya x -> x0-x.

        changeVarExpressionSo = sollya_lib_build_function_sub( \

                              sollya_lib_build_function_free_variable(), \

                              sollya_lib_copy_obj(intervalCenterSo))

        polyVarChangedSo = sollya_lib_evaluate(polySo, changeVarExpressionSo)

        sollya_lib_clear_obj(changeVarExpressionSo)

        resultArray.append((polySo, polyVarChangedSo, boundsSo, \

                            intervalCenterSo, maxErrorSo))

        if internalSollyaPrecSa != currentSollyaPrecSa:

            pobyso_set_prec_sa_so(currentSollyaPrecSa)

        sollya_lib_clear_obj(currentSollyaPrecSo)

        sollya_lib_clear_obj(functionSo)

        sollya_lib_clear_obj(degreeSo)

        sollya_lib_clear_obj(scaledBoundsSo)

        #print "Approximation error:", errorSa

        return resultArray

    # Compute the next upper bound.

    # The following ratio is always >= 1

    currentErrorRatio = approxPrecSa / errorSa

    # Starting point for the next upper bound.

    currentScaledUpperBoundSa = boundsSa.endpoints()[1]

    boundsWidthSa = boundsSa.endpoints()[1] - boundsSa.endpoints()[0]

    # Compute the increment.

    if currentErrorRatio > RR('1000'): # ]1.5, infinity[

        currentScaledUpperBoundSa += \

                        currentErrorRatio * boundsWidthSa * 2

    else:  # [1, 1.5]

        currentScaledUpperBoundSa += \

                        (RR('1.0') + currentErrorRatio.log() / 500) * boundsWidthSa

    # Take into account the original interval upper bound.

    if currentScaledUpperBoundSa > scaledUpperBoundSa:

        currentScaledUpperBoundSa = scaledUpperBoundSa

    # Compute the other expansions.

    while boundsSa.endpoints()[1] < scaledUpperBoundSa:

        currentScaledLowerBoundSa = boundsSa.endpoints()[1]

        (polySo, boundsSo, intervalCenterSo, maxErrorSo) = \

            slz_compute_polynomial_and_interval(functionSo, degreeSo,

                                            currentScaledLowerBoundSa,

                                            currentScaledUpperBoundSa,

                                            approxPrecSa,

                                            internalSollyaPrecSa)

        errorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

        if  errorSa < approxPrecSa:

            # Change variable stuff

            #print "Approximation error:", errorSa

            changeVarExpressionSo = sollya_lib_build_function_sub(

                                    sollya_lib_build_function_free_variable(),

                                    sollya_lib_copy_obj(intervalCenterSo))

            polyVarChangedSo = sollya_lib_evaluate(polySo, changeVarExpressionSo)

            sollya_lib_clear_obj(changeVarExpressionSo)

            resultArray.append((polySo, polyVarChangedSo, boundsSo, \

                                intervalCenterSo, maxErrorSo))

        boundsSa = pobyso_range_to_interval_so_sa(boundsSo, realIntervalField)

        # Compute the next upper bound.

        # The following ratio is always >= 1

        currentErrorRatio = approxPrecSa / errorSa

        # Starting point for the next upper bound.

        currentScaledUpperBoundSa = boundsSa.endpoints()[1]

        boundsWidthSa = boundsSa.endpoints()[1] - boundsSa.endpoints()[0]

        # Compute the increment.

        if currentErrorRatio > RR('1000'): # ]1.5, infinity[

            currentScaledUpperBoundSa += \

                            currentErrorRatio * boundsWidthSa * 2

        else:  # [1, 1.5]

            currentScaledUpperBoundSa += \

                            (RR('1.0') + currentErrorRatio.log()/500) * boundsWidthSa

        #print "currentErrorRatio:", currentErrorRatio

        #print "currentScaledUpperBoundSa", currentScaledUpperBoundSa

        # Test for insufficient precision.

        if currentScaledUpperBoundSa == scaledLowerBoundSa:

            print "Can't shrink the interval anymore!"

            print "You should consider increasing the Sollya internal precision"

            print "or the polynomial degree."

            print "Giving up!"

            if internalSollyaPrecSa != currentSollyaPrecSa:

                pobyso_set_prec_sa_so(currentSollyaPrecSa)

            sollya_lib_clear_obj(currentSollyaPrecSo)

            sollya_lib_clear_obj(functionSo)

            sollya_lib_clear_obj(degreeSo)

            sollya_lib_clear_obj(scaledBoundsSo)

            return None

        if currentScaledUpperBoundSa > scaledUpperBoundSa:

            currentScaledUpperBoundSa = scaledUpperBoundSa

    if internalSollyaPrecSa > currentSollyaPrecSa:

        pobyso_set_prec_so_so(currentSollyaPrecSo)

    sollya_lib_clear_obj(currentSollyaPrecSo)

    sollya_lib_clear_obj(functionSo)

    sollya_lib_clear_obj(degreeSo)

    sollya_lib_clear_obj(scaledBoundsSo)

    return(resultArray)

# End slz_get_intervals_and_polynomials

def slz_interval_scaling_expression(boundsInterval, expVar):

    """

    Compute the scaling expression to map an interval that span at most

    a single binade to [1, 2) and the inverse expression as well.

    Not very sure that the transformation makes sense for negative numbers.

    """

    # The scaling offset is only used for negative numbers.

    if abs(boundsInterval.endpoints()[0]) < 1:

        if boundsInterval.endpoints()[0] >= 0:

            scalingCoeff = 2^floor(boundsInterval.endpoints()[0].log2())

            invScalingCoeff = 1/scalingCoeff

            return((scalingCoeff * expVar,

                    invScalingCoeff * expVar))

        else:

            scalingCoeff = \

                2^(floor((-boundsInterval.endpoints()[0]).log2()) - 1)

            scalingOffset = -3 * scalingCoeff

            return((scalingCoeff * expVar + scalingOffset,

                    1/scalingCoeff * expVar + 3))

    else:

        if boundsInterval.endpoints()[0] >= 0:

            scalingCoeff = 2^floor(boundsInterval.endpoints()[0].log2())

            scalingOffset = 0

            return((scalingCoeff * expVar,

                    1/scalingCoeff * expVar))

        else:

            scalingCoeff = \

                2^(floor((-boundsInterval.endpoints()[1]).log2()))

            scalingOffset =  -3 * scalingCoeff

            #scalingOffset = 0

            return((scalingCoeff * expVar + scalingOffset,

                    1/scalingCoeff * expVar + 3))

def slz_interval_and_polynomial_to_sage(polyRangeCenterErrorSo):

    """

    Compute the Sage version of the Taylor polynomial and it's

    companion data (interval, center...)

    The input parameter is a five elements tuple:

    - [0]: the polyomial (without variable change), as polynomial over a

           real ring;

    - [1]: the polyomial (with variable change done in Sollya), as polynomial

           over a real ring;

    - [2]: the interval (as Sollya range);

    - [3]: the interval center;

    - [4]: the approximation error.

    The function return a 5 elements tuple: formed with all the

    input elements converted into their Sollya counterpart.

    """

    polynomialSa = pobyso_get_poly_so_sa(polyRangeCenterErrorSo[0])

    polynomialChangedVarSa = pobyso_get_poly_so_sa(polyRangeCenterErrorSo[1])

    intervalSa = \

            pobyso_get_interval_from_range_so_sa(polyRangeCenterErrorSo[2])

    centerSa = \

            pobyso_get_constant_as_rn_with_rf_so_sa(polyRangeCenterErrorSo[3])

    errorSa = \

            pobyso_get_constant_as_rn_with_rf_so_sa(polyRangeCenterErrorSo[4])

    return((polynomialSa, polynomialChangedVarSa, intervalSa, centerSa, errorSa))

    # End slz_interval_and_polynomial_to_sage

def slz_rat_poly_of_int_to_poly_for_coppersmith(ratPolyOfInt,

                                                precision,

                                                targetHardnessToRound,

                                                variable1,

                                                variable2):

    """

    Creates a new multivariate polynomial with integer coefficients for use

     with the Coppersmith method.

    A the same time it computes :

    - 2^K (N);

    - 2^k (bound on the second variable)

    - lcm

    :param ratPolyOfInt: a polynomial with rational coefficients and integer

                         variables.

    :param precision: the precision of the floating-point coefficients.

    :param targetHardnessToRound: the hardness to round we want to check.

    :param variable1: the first variable of the polynomial (an expression).

    :param variable2: the second variable of the polynomial (an expression).

    :returns: a 4 elements tuple:

                - the polynomial;

                - the modulus (N);

                - the t bound;

                - the lcm used to compute the integral coefficients and the

                  module.

    """

    # Create a new integer polynomial ring.

    IP = PolynomialRing(ZZ, name=str(variable1) + "," + str(variable2))

    # Coefficients are issued in the increasing power order.

    ratPolyCoefficients = ratPolyOfInt.coefficients()

    # Print the reversed list for debugging.

    print "Rational polynomial coefficients:", ratPolyCoefficients[::-1]

    # Build the list of number we compute the lcm of.

    coefficientDenominators = sro_denominators(ratPolyCoefficients)

    coefficientDenominators.append(2^precision)

    coefficientDenominators.append(2^(targetHardnessToRound + 1))

    leastCommonMultiple = lcm(coefficientDenominators)

    # Compute the expression corresponding to the new polynomial

    coefficientNumerators =  sro_numerators(ratPolyCoefficients)

    #print coefficientNumerators

    polynomialExpression = 0

    power = 0

    # Iterate over two lists at the same time, stop when the shorter is

    # exhausted.

    for numerator, denominator in \

                        zip(coefficientNumerators, coefficientDenominators):

        multiplicator = leastCommonMultiple / denominator

        newCoefficient = numerator * multiplicator

        polynomialExpression += newCoefficient * variable1^power

        power +=1

    polynomialExpression += - variable2

    return (IP(polynomialExpression),

            leastCommonMultiple / 2^precision, # 2^K or N.

            leastCommonMultiple / 2^(targetHardnessToRound + 1), # tBound

            leastCommonMultiple) # If we want to make test computations.

# End slz_ratPoly_of_int_to_poly_for_coppersmith

def slz_rat_poly_of_rat_to_rat_poly_of_int(ratPolyOfRat,

                                          precision):

    """

    Makes a variable substitution into the input polynomial so that the output

    polynomial can take integer arguments.

    All variables of the input polynomial "have precision p". That is to say

    that they are rationals with denominator == 2^(precision - 1):

    x = y/2^(precision - 1).

    We "incorporate" these denominators into the coefficients with,

    respectively, the "right" power.

    """

    polynomialField = ratPolyOfRat.parent()

    polynomialVariable = ratPolyOfRat.variables()[0]

    #print "The polynomial field is:", polynomialField

    return \

        polynomialField(ratPolyOfRat.subs({polynomialVariable : \

                                   polynomialVariable/2^(precision-1)}))

    # Return a tuple:

    # - the bivariate integer polynomial in (i,j);

    # - 2^K

# End slz_rat_poly_of_rat_to_rat_poly_of_int

print "\t...sageSLZ loaded"