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/*
*
* * Copyright 2015 Skymind,Inc.
* *
* * Licensed under the Apache License, Version 2.0 (the "License");
* * you may not use this file except in compliance with the License.
* * You may obtain a copy of the License at
* *
* * http://www.apache.org/licenses/LICENSE-2.0
* *
* * Unless required by applicable law or agreed to in writing, software
* * distributed under the License is distributed on an "AS IS" BASIS,
* * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* * See the License for the specific language governing permissions and
* * limitations under the License.
*
*
*/
package org.nd4j.linalg.util;
// Code
import org.apache.commons.math3.util.FastMath;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.MathContext;
import java.security.ProviderException;
/**
* BigDecimal special functions.
*/
public class BigDecimalMath {
/**
* The base of the natural logarithm in a predefined accuracy.
* \protect\vrule width0pt\protect\href{http://www.cs.arizona.edu/icon/oddsends/e.htm}{http://www.cs.arizona.edu/icon/oddsends/e.htm}
* The precision of the predefined constant is one less than
* the string’s length, taking into account the decimal dot.
* static int E_PRECISION = E.length()-1 ;
*/
static BigDecimal E = new BigDecimal("2.71828182845904523536028747135266249775724709369995957496696762772407663035354"
+ "759457138217852516642742746639193200305992181741359662904357290033429526059563"
+ "073813232862794349076323382988075319525101901157383418793070215408914993488416"
+ "750924476146066808226480016847741185374234544243710753907774499206955170276183"
+ "860626133138458300075204493382656029760673711320070932870912744374704723069697"
+ "720931014169283681902551510865746377211125238978442505695369677078544996996794"
+ "686445490598793163688923009879312773617821542499922957635148220826989519366803"
+ "318252886939849646510582093923982948879332036250944311730123819706841614039701"
+ "983767932068328237646480429531180232878250981945581530175671736133206981125099"
+ "618188159304169035159888851934580727386673858942287922849989208680582574927961"
+ "048419844436346324496848756023362482704197862320900216099023530436994184914631"
+ "409343173814364054625315209618369088870701676839642437814059271456354906130310"
+ "720851038375051011574770417189861068739696552126715468895703503540212340784981"
+ "933432106817012100562788023519303322474501585390473041995777709350366041699732"
+ "972508868769664035557071622684471625607988265178713419512466520103059212366771"
+ "943252786753985589448969709640975459185695638023637016211204774272283648961342"
+ "251644507818244235294863637214174023889344124796357437026375529444833799801612"
+ "549227850925778256209262264832627793338656648162772516401910590049164499828931");
/**
* Euler’s constant Pi.
* \protect\vrule width0pt\protect\href{http://www.cs.arizona.edu/icon/oddsends/pi.htm}{http://www.cs.arizona.edu/icon/oddsends/pi.htm}
*/
static BigDecimal PI = new BigDecimal("3.14159265358979323846264338327950288419716939937510582097494459230781640628620"
+ "899862803482534211706798214808651328230664709384460955058223172535940812848111"
+ "745028410270193852110555964462294895493038196442881097566593344612847564823378"
+ "678316527120190914564856692346034861045432664821339360726024914127372458700660"
+ "631558817488152092096282925409171536436789259036001133053054882046652138414695"
+ "194151160943305727036575959195309218611738193261179310511854807446237996274956"
+ "735188575272489122793818301194912983367336244065664308602139494639522473719070"
+ "217986094370277053921717629317675238467481846766940513200056812714526356082778"
+ "577134275778960917363717872146844090122495343014654958537105079227968925892354"
+ "201995611212902196086403441815981362977477130996051870721134999999837297804995"
+ "105973173281609631859502445945534690830264252230825334468503526193118817101000"
+ "313783875288658753320838142061717766914730359825349042875546873115956286388235"
+ "378759375195778185778053217122680661300192787661119590921642019893809525720106"
+ "548586327886593615338182796823030195203530185296899577362259941389124972177528"
+ "347913151557485724245415069595082953311686172785588907509838175463746493931925"
+ "506040092770167113900984882401285836160356370766010471018194295559619894676783"
+ "744944825537977472684710404753464620804668425906949129331367702898915210475216"
+ "205696602405803815019351125338243003558764024749647326391419927260426992279678"
+ "235478163600934172164121992458631503028618297455570674983850549458858692699569"
+ "092721079750930295532116534498720275596023648066549911988183479775356636980742"
+ "654252786255181841757467289097777279380008164706001614524919217321721477235014");
/**
* Euler-Mascheroni constant lower-case gamma.
* 13
* \protect\vrule width0pt\protect\href{http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap35.html}{http:/
*/
static BigDecimal GAMMA = new BigDecimal("0.577215664901532860606512090082402431"
+ "0421593359399235988057672348848677267776646709369470632917467495146314472498070"
+ "8248096050401448654283622417399764492353625350033374293733773767394279259525824"
+ "7094916008735203948165670853233151776611528621199501507984793745085705740029921"
+ "3547861466940296043254215190587755352673313992540129674205137541395491116851028"
+ "0798423487758720503843109399736137255306088933126760017247953783675927135157722"
+ "6102734929139407984301034177717780881549570661075010161916633401522789358679654"
+ "9725203621287922655595366962817638879272680132431010476505963703947394957638906"
+ "5729679296010090151251959509222435014093498712282479497471956469763185066761290"
+ "6381105182419744486783638086174945516989279230187739107294578155431600500218284"
+ "4096053772434203285478367015177394398700302370339518328690001558193988042707411"
+ "5422278197165230110735658339673487176504919418123000406546931429992977795693031"
+ "0050308630341856980323108369164002589297089098548682577736428825395492587362959"
+ "6133298574739302373438847070370284412920166417850248733379080562754998434590761"
+ "6431671031467107223700218107450444186647591348036690255324586254422253451813879"
+ "1243457350136129778227828814894590986384600629316947188714958752549236649352047"
+ "3243641097268276160877595088095126208404544477992299157248292516251278427659657"
+ "0832146102982146179519579590959227042089896279712553632179488737642106606070659"
+ "8256199010288075612519913751167821764361905705844078357350158005607745793421314"
+ "49885007864151716151945");
/**
* Natural logarithm of 2.
* \protect\vrule width0pt\protect\href{http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap58.html}{http:/
*/
static BigDecimal LOG2 = new BigDecimal("0.693147180559945309417232121458176568075"
+ "50013436025525412068000949339362196969471560586332699641868754200148102057068573"
+ "368552023575813055703267075163507596193072757082837143519030703862389167347112335"
+ "011536449795523912047517268157493206515552473413952588295045300709532636664265410"
+ "423915781495204374043038550080194417064167151864471283996817178454695702627163106"
+ "454615025720740248163777338963855069526066834113727387372292895649354702576265209"
+ "885969320196505855476470330679365443254763274495125040606943814710468994650622016"
+ "772042452452961268794654619316517468139267250410380254625965686914419287160829380"
+ "317271436778265487756648508567407764845146443994046142260319309673540257444607030"
+ "809608504748663852313818167675143866747664789088143714198549423151997354880375165"
+ "861275352916610007105355824987941472950929311389715599820565439287170007218085761"
+ "025236889213244971389320378439353088774825970171559107088236836275898425891853530"
+ "243634214367061189236789192372314672321720534016492568727477823445353476481149418"
+ "642386776774406069562657379600867076257199184734022651462837904883062033061144630"
+ "073719489002743643965002580936519443041191150608094879306786515887090060520346842"
+ "973619384128965255653968602219412292420757432175748909770675268711581705113700915"
+ "894266547859596489065305846025866838294002283300538207400567705304678700184162404"
+ "418833232798386349001563121889560650553151272199398332030751408426091479001265168"
+ "243443893572472788205486271552741877243002489794540196187233980860831664811490930"
+ "667519339312890431641370681397776498176974868903887789991296503619270710889264105"
+ "230924783917373501229842420499568935992206602204654941510613");
/**
* A suggestion for the maximum numter of terms in the Taylor expansion of the exponential.
*/
static private int TAYLOR_NTERM = 8;
/**
* Euler’s constant.
*
* @param mc The required precision of the result.
* @return 3.14159...
*/
static public BigDecimal pi(final MathContext mc) {
/* look it up if possible */
if (mc.getPrecision() < PI.precision()) {
return PI.round(mc);
} else {
/* Broadhurst \protect\vrule width0pt\protect\href{http://arxiv.org/abs/math/9803067}{arXiv:math/9803067}
*/
int[] a = {1, 0, 0, -1, -1, -1, 0, 0};
BigDecimal S = broadhurstBBP(1, 1, a, mc);
return multiplyRound(S, 8);
}
} /* BigDecimalMath.pi */
/**
* Euler-Mascheroni constant.
*
* @param mc The required precision of the result.
* @return 0.577...
*/
static public BigDecimal gamma(MathContext mc) {
/* look it up if possible */
if (mc.getPrecision() < GAMMA.precision()) {
return GAMMA.round(mc);
} else {
double eps = prec2err(0.577, mc.getPrecision());
/* Euler-Stieltjes as shown in Dilcher, Aequat Math 48 (1) (1994) 55-85
14
*/
MathContext mcloc = new MathContext(2 + mc.getPrecision());
BigDecimal resul = BigDecimal.ONE;
resul = resul.add(log(2, mcloc));
resul = resul.subtract(log(3, mcloc));
/* how many terms: zeta-1 falls as 1/2^(2n+1), so the
* terms drop faster than 1/2^(4n+2). Set 1/2^(4kmax+2) < eps.
* Leading term zeta(3)/(4^1*3) is 0.017. Leading zeta(3) is 1.2. Log(2) is 0.7
*/
int kmax = (int) ((Math.log(eps / 0.7) - 2.) / 4.);
mcloc = new MathContext(1 + err2prec(1.2, eps / kmax));
for (int n = 1; ; n++) {
/* zeta is close to 1. Division of zeta-1 through
* 4^n*(2n+1) means divion through roughly 2^(2n+1)
*/
BigDecimal c = zeta(2 * n + 1, mcloc).subtract(BigDecimal.ONE);
BigInteger fourn = new BigInteger("" + (2 * n + 1));
fourn = fourn.shiftLeft(2 * n);
c = divideRound(c, fourn);
resul = resul.subtract(c);
if (c.doubleValue() < 0.1 * eps) {
break;
}
}
return resul.round(mc);
}
} /* BigDecimalMath.gamma */
/**
* The square root.
*
* @param x the non-negative argument.
* @return the square root of the BigDecimal rounded to the precision implied by x.
*/
static public BigDecimal sqrt(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
throw new ArithmeticException("negative argument " + x.toString() + " of square root");
}
return root(2, x);
} /* BigDecimalMath.sqrt */
/**
* The cube root.
*
* @param x The argument.
* @return The cubic root of the BigDecimal rounded to the precision implied by x.
* The sign of the result is the sign of the argument.
*/
static public BigDecimal cbrt(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return root(3, x.negate()).negate();
} else {
return root(3, x);
}
} /* BigDecimalMath.cbrt */
/**
* The integer root.
*
* @param n the positive argument.
* @param x the non-negative argument.
* @return The n-th root of the BigDecimal rounded to the precision implied by x, x^(1/n).
*/
static public BigDecimal root(final int n, final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
throw new ArithmeticException("negative argument " + x.toString() + " of root");
}
if (n <= 0) {
throw new ArithmeticException("negative power " + n + " of root");
}
if (n == 1) {
return x;
}
/* start the computation from a double precision estimate */
BigDecimal s = new BigDecimal(Math.pow(x.doubleValue(), 1.0 / n));
/* this creates nth with nominal precision of 1 digit
*/
final BigDecimal nth = new BigDecimal(n);
/* Specify an internal accuracy within the loop which is
* slightly larger than what is demanded by ’eps’ below.
*/
final BigDecimal xhighpr = scalePrec(x, 2);
MathContext mc = new MathContext(2 + x.precision());
/* Relative accuracy of the result is eps.
*/
final double eps = x.ulp().doubleValue() / (2 * n * x.doubleValue());
for (; ; ) {
/* s = s -(s/n-x/n/s^(n-1)) = s-(s-x/s^(n-1))/n; test correction s/n-x/s for being
* smaller than the precision requested. The relative correction is (1-x/s^n)/n,
*/
BigDecimal c = xhighpr.divide(s.pow(n - 1), mc);
c = s.subtract(c);
MathContext locmc = new MathContext(c.precision());
c = c.divide(nth, locmc);
s = s.subtract(c);
if (Math.abs(c.doubleValue() / s.doubleValue()) < eps) {
break;
}
}
return s.round(new MathContext(err2prec(eps)));
} /* BigDecimalMath.root */
/**
* The hypotenuse.
*
* @param x the first argument.
* @param y the second argument.
* @return the square root of the sum of the squares of the two arguments, sqrt(x^2+y^2).
*/
static public BigDecimal hypot(final BigDecimal x, final BigDecimal y) {
/* compute x^2+y^2
*/
BigDecimal z = x.pow(2).add(y.pow(2));
/* truncate to the precision set by x and y. Absolute error = 2*x*xerr+2*y*yerr,
* where the two errors are 1/2 of the ulp’s. Two intermediate protectio digits.
*/
BigDecimal zerr = x.abs().multiply(x.ulp()).add(y.abs().multiply(y.ulp()));
MathContext mc = new MathContext(2 + err2prec(z, zerr));
/* Pull square root */
z = sqrt(z.round(mc));
/* Final rounding. Absolute error in the square root is (y*yerr+x*xerr)/z, where zerr holds 2*(x*xerr+y*yerr).
*/
mc = new MathContext(err2prec(z.doubleValue(), 0.5 * zerr.doubleValue() / z.doubleValue()));
return z.round(mc);
} /* BigDecimalMath.hypot */
/**
* The hypotenuse.
*
* @param n the first argument.
* @param x the second argument.
* @return the square root of the sum of the squares of the two arguments, sqrt(n^2+x^2).
*/
static public BigDecimal hypot(final int n, final BigDecimal x) {
/* compute n^2+x^2 in infinite precision
*/
BigDecimal z = (new BigDecimal(n)).pow(2).add(x.pow(2));
/* Truncate to the precision set by x. Absolute error = in z (square of the result) is |2*x*xerr|,
* where the error is 1/2 of the ulp. Two intermediate protection digits.
* zerr is a signed value, but used only in conjunction with err2prec(), so this feature does not harm.
*/
double zerr = x.doubleValue() * x.ulp().doubleValue();
MathContext mc = new MathContext(2 + err2prec(z.doubleValue(), zerr));
/* Pull square root */
z = sqrt(z.round(mc));
/* Final rounding. Absolute error in the square root is x*xerr/z, where zerr holds 2*x*xerr.
*/
mc = new MathContext(err2prec(z.doubleValue(), 0.5 * zerr / z.doubleValue()));
return z.round(mc);
} /* BigDecimalMath.hypot */
/**
* The exponential function.
*
* @param x the argument.
* @return exp(x).
* The precision of the result is implicitly defined by the precision in the argument.
* 16
* In particular this means that "Invalid Operation" errors are thrown if catastrophic
* cancellation of digits causes the result to have no valid digits left.
*/
static public BigDecimal exp(BigDecimal x) {
/* To calculate the value if x is negative, use exp(-x) = 1/exp(x)
*/
if (x.compareTo(BigDecimal.ZERO) < 0) {
final BigDecimal invx = exp(x.negate());
/* Relative error in inverse of invx is the same as the relative errror in invx.
* This is used to define the precision of the result.
*/
MathContext mc = new MathContext(invx.precision());
return BigDecimal.ONE.divide(invx, mc);
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
/* recover the valid number of digits from x.ulp(), if x hits the
* zero. The x.precision() is 1 then, and does not provide this information.
*/
return scalePrec(BigDecimal.ONE, -(int) (Math.log10(x.ulp().doubleValue())));
} else {
/* Push the number in the Taylor expansion down to a small
* value where TAYLOR_NTERM terms will do. If x<1, the n-th term is of the order
* x^n/n!, and equal to both the absolute and relative error of the result
* since the result is close to 1. The x.ulp() sets the relative and absolute error
* of the result, as estimated from the first Taylor term.
* We want x^TAYLOR_NTERM/TAYLOR_NTERM! < x.ulp, which is guaranteed if
* x^TAYLOR_NTERM < TAYLOR_NTERM*(TAYLOR_NTERM-1)*...*x.ulp.
*/
final double xDbl = x.doubleValue();
final double xUlpDbl = x.ulp().doubleValue();
if (Math.pow(xDbl, TAYLOR_NTERM) < TAYLOR_NTERM * (TAYLOR_NTERM - 1.0) * (TAYLOR_NTERM - 2.0) * xUlpDbl) {
/* Add TAYLOR_NTERM terms of the Taylor expansion (Euler’s sum formula)
*/
BigDecimal resul = BigDecimal.ONE;
/* x^i */
BigDecimal xpowi = BigDecimal.ONE;
/* i factorial */
BigInteger ifac = BigInteger.ONE;
/* TAYLOR_NTERM terms to be added means we move x.ulp() to the right
* for each power of 10 in TAYLOR_NTERM, so the addition won’t add noise beyond
* what’s already in x.
*/
MathContext mcTay = new MathContext(err2prec(1., xUlpDbl / TAYLOR_NTERM));
for (int i = 1; i <= TAYLOR_NTERM; i++) {
ifac = ifac.multiply(new BigInteger("" + i));
xpowi = xpowi.multiply(x);
final BigDecimal c = xpowi.divide(new BigDecimal(ifac), mcTay);
resul = resul.add(c);
if (Math.abs(xpowi.doubleValue()) < i && Math.abs(c.doubleValue()) < 0.5 * xUlpDbl) {
break;
}
}
/* exp(x+deltax) = exp(x)(1+deltax) if deltax is <<1. So the relative error
* in the result equals the absolute error in the argument.
*/
MathContext mc = new MathContext(err2prec(xUlpDbl / 2.));
return resul.round(mc);
} else {
/* Compute exp(x) = (exp(0.1*x))^10. Division by 10 does not lead
* to loss of accuracy.
*/
int exSc = (int) (1.0 - Math.log10(TAYLOR_NTERM * (TAYLOR_NTERM - 1.0) * (TAYLOR_NTERM - 2.0) * xUlpDbl
/ Math.pow(xDbl, TAYLOR_NTERM)) / (TAYLOR_NTERM - 1.0));
BigDecimal xby10 = x.scaleByPowerOfTen(-exSc);
BigDecimal expxby10 = exp(xby10);
/* Final powering by 10 means that the relative error of the result
* is 10 times the relative error of the base (First order binomial expansion).
* This looses one digit.
*/
MathContext mc = new MathContext(expxby10.precision() - exSc);
/* Rescaling the powers of 10 is done in chunks of a maximum of 8 to avoid an invalid operation
17
* response by the BigDecimal.pow library or integer overflow.
*/
while (exSc > 0) {
int exsub = Math.min(8, exSc);
exSc -= exsub;
MathContext mctmp = new MathContext(expxby10.precision() - exsub + 2);
int pex = 1;
while (exsub-- > 0) {
pex *= 10;
}
expxby10 = expxby10.pow(pex, mctmp);
}
return expxby10.round(mc);
}
}
} /* BigDecimalMath.exp */
/**
* The base of the natural logarithm.
*
* @param mc the required precision of the result
* @return exp(1) = 2.71828....
*/
static public BigDecimal exp(final MathContext mc) {
/* look it up if possible */
if (mc.getPrecision() < E.precision()) {
return E.round(mc);
} else {
/* Instantiate a 1.0 with the requested pseudo-accuracy
* and delegate the computation to the public method above.
*/
BigDecimal uni = scalePrec(BigDecimal.ONE, mc.getPrecision());
return exp(uni);
}
} /* BigDecimalMath.exp */
/**
* The natural logarithm.
*
* @param x the argument.
* @return ln(x).
* The precision of the result is implicitly defined by the precision in the argument.
*/
static public BigDecimal log(BigDecimal x) {
/* the value is undefined if x is negative.
*/
if (x.compareTo(BigDecimal.ZERO) < 0) {
throw new ArithmeticException("Cannot take log of negative " + x.toString());
} else if (x.compareTo(BigDecimal.ONE) == 0) {
/* log 1. = 0. */
return scalePrec(BigDecimal.ZERO, x.precision() - 1);
} else if (Math.abs(x.doubleValue() - 1.0) <= 0.3) {
/* The standard Taylor series around x=1, z=0, z=x-1. Abramowitz-Stegun 4.124.
* The absolute error is err(z)/(1+z) = err(x)/x.
*/
BigDecimal z = scalePrec(x.subtract(BigDecimal.ONE), 2);
BigDecimal zpown = z;
double eps = 0.5 * x.ulp().doubleValue() / Math.abs(x.doubleValue());
BigDecimal resul = z;
for (int k = 2; ; k++) {
zpown = multiplyRound(zpown, z);
BigDecimal c = divideRound(zpown, k);
if (k % 2 == 0) {
resul = resul.subtract(c);
} else {
resul = resul.add(c);
}
if (Math.abs(c.doubleValue()) < eps) {
break;
}
}
MathContext mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
} else {
final double xDbl = x.doubleValue();
final double xUlpDbl = x.ulp().doubleValue();
/* Map log(x) = log root[r](x)^r = r*log( root[r](x)) with the aim
* to move roor[r](x) near to 1.2 (that is, below the 0.3 appearing above), where log(1.2) is roughly 0.2.
*/
int r = (int) (Math.log(xDbl) / 0.2);
/* Since the actual requirement is a function of the value 0.3 appearing above,
* we avoid the hypothetical case of endless recurrence by ensuring that r >= 2.
*/
r = Math.max(2, r);
/* Compute r-th root with 2 additional digits of precision
*/
BigDecimal xhighpr = scalePrec(x, 2);
BigDecimal resul = root(r, xhighpr);
resul = log(resul).multiply(new BigDecimal(r));
/* error propagation: log(x+errx) = log(x)+errx/x, so the absolute error
* in the result equals the relative error in the input, xUlpDbl/xDbl .
*/
MathContext mc = new MathContext(err2prec(resul.doubleValue(), xUlpDbl / xDbl));
return resul.round(mc);
}
} /* BigDecimalMath.log */
/**
* The natural logarithm.
*
* @param n The main argument, a strictly positive integer.
* @param mc The requirements on the precision.
* @return ln(n).
*/
static public BigDecimal log(int n, final MathContext mc) {
/* the value is undefined if x is negative.
*/
if (n <= 0) {
throw new ArithmeticException("Cannot take log of negative " + n);
} else if (n == 1) {
return BigDecimal.ZERO;
} else if (n == 2) {
if (mc.getPrecision() < LOG2.precision()) {
return LOG2.round(mc);
} else {
/* Broadhurst \protect\vrule width0pt\protect\href{http://arxiv.org/abs/math/9803067}{arXiv:math/9803067}
* Error propagation: the error in log(2) is twice the error in S(2,-5,...).
*/
int[] a = {2, -5, -2, -7, -2, -5, 2, -3};
BigDecimal S = broadhurstBBP(2, 1, a, new MathContext(1 + mc.getPrecision()));
S = S.multiply(new BigDecimal(8));
S = sqrt(divideRound(S, 3));
return S.round(mc);
}
} else if (n == 3) {
/* summation of a series roughly proportional to (7/500)^k. Estimate count
* of terms to estimate the precision (drop the favorable additional
* 1/k here): 0.013^k <= 10^(-precision), so k*log10(0.013) <= -precision
* so k>= precision/1.87.
*/
int kmax = (int) (mc.getPrecision() / 1.87);
MathContext mcloc = new MathContext(mc.getPrecision() + 1 + (int) (Math.log10(kmax * 0.693 / 1.098)));
BigDecimal log3 = multiplyRound(log(2, mcloc), 19);
/* log3 is roughly 1, so absolute and relative error are the same. The
* result will be divided by 12, so a conservative error is the one
* already found in mc
*/
double eps = prec2err(1.098, mc.getPrecision()) / kmax;
Rational r = new Rational(7153, 524288);
Rational pk = new Rational(7153, 524288);
for (int k = 1; ; k++) {
Rational tmp = pk.divide(k);
if (tmp.doubleValue() < eps) {
break;
}
/* how many digits of tmp do we need in the sum?
*/
mcloc = new MathContext(err2prec(tmp.doubleValue(), eps));
BigDecimal c = pk.divide(k).BigDecimalValue(mcloc);
if (k % 2 != 0) {
log3 = log3.add(c);
} else {
log3 = log3.subtract(c);
}
pk = pk.multiply(r);
}
log3 = divideRound(log3, 12);
return log3.round(mc);
} else if (n == 5) {
/* summation of a series roughly proportional to (7/160)^k. Estimate count
* of terms to estimate the precision (drop the favorable additional
* 1/k here): 0.046^k <= 10^(-precision), so k*log10(0.046) <= -precision
* so k>= precision/1.33.
*/
int kmax = (int) (mc.getPrecision() / 1.33);
MathContext mcloc = new MathContext(mc.getPrecision() + 1 + (int) (Math.log10(kmax * 0.693 / 1.609)));
BigDecimal log5 = multiplyRound(log(2, mcloc), 14);
/* log5 is roughly 1.6, so absolute and relative error are the same. The
* result will be divided by 6, so a conservative error is the one
* already found in mc
*/
double eps = prec2err(1.6, mc.getPrecision()) / kmax;
Rational r = new Rational(759, 16384);
Rational pk = new Rational(759, 16384);
for (int k = 1; ; k++) {
Rational tmp = pk.divide(k);
if (tmp.doubleValue() < eps) {
break;
}
/* how many digits of tmp do we need in the sum?
*/
mcloc = new MathContext(err2prec(tmp.doubleValue(), eps));
BigDecimal c = pk.divide(k).BigDecimalValue(mcloc);
log5 = log5.subtract(c);
pk = pk.multiply(r);
}
log5 = divideRound(log5, 6);
return log5.round(mc);
} else if (n == 7) {
/* summation of a series roughly proportional to (1/8)^k. Estimate count
* of terms to estimate the precision (drop the favorable additional
* 1/k here): 0.125^k <= 10^(-precision), so k*log10(0.125) <= -precision
* so k>= precision/0.903.
*/
int kmax = (int) (mc.getPrecision() / 0.903);
MathContext mcloc = new MathContext(mc.getPrecision() + 1 + (int) (Math.log10(kmax * 3 * 0.693 / 1.098)));
BigDecimal log7 = multiplyRound(log(2, mcloc), 3);
/* log7 is roughly 1.9, so absolute and relative error are the same.
*/
double eps = prec2err(1.9, mc.getPrecision()) / kmax;
Rational r = new Rational(1, 8);
Rational pk = new Rational(1, 8);
for (int k = 1; ; k++) {
Rational tmp = pk.divide(k);
if (tmp.doubleValue() < eps) {
break;
}
/* how many digits of tmp do we need in the sum?
*/
mcloc = new MathContext(err2prec(tmp.doubleValue(), eps));
BigDecimal c = pk.divide(k).BigDecimalValue(mcloc);
log7 = log7.subtract(c);
pk = pk.multiply(r);
}
return log7.round(mc);
} else {
/* At this point one could either forward to the log(BigDecimal) signature (implemented)
* or decompose n into Ifactors and use an implemenation of all the prime bases.
* Estimate of the result; convert the mc argument to an absolute error eps
* log(n+errn) = log(n)+errn/n = log(n)+eps
*/
double res = Math.log((double) n);
double eps = prec2err(res, mc.getPrecision());
/* errn = eps*n, convert absolute error in result to requirement on absolute error in input
*/
eps *= n;
/* Convert this absolute requirement of error in n to a relative error in n
*/
final MathContext mcloc = new MathContext(1 + err2prec((double) n, eps));
/* Padd n with a number of zeros to trigger the required accuracy in
* the standard signature method
*/
BigDecimal nb = scalePrec(new BigDecimal(n), mcloc);
return log(nb);
}
} /* log */
/**
* The natural logarithm.
*
* @param r The main argument, a strictly positive value.
* @param mc The requirements on the precision.
* @return ln(r).
*/
static public BigDecimal log(final Rational r, final MathContext mc) {
/* the value is undefined if x is negative.
*/
if (r.compareTo(Rational.ZERO) <= 0) {
throw new ArithmeticException("Cannot take log of negative " + r.toString());
} else if (r.compareTo(Rational.ONE) == 0) {
return BigDecimal.ZERO;
} else {
/* log(r+epsr) = log(r)+epsr/r. Convert the precision to an absolute error in the result.
* eps contains the required absolute error of the result, epsr/r.
*/
double eps = prec2err(Math.log(r.doubleValue()), mc.getPrecision());
/* Convert this further into a requirement of the relative precision in r, given that
* epsr/r is also the relative precision of r. Add one safety digit.
*/
MathContext mcloc = new MathContext(1 + err2prec(eps));
final BigDecimal resul = log(r.BigDecimalValue(mcloc));
return resul.round(mc);
}
} /* log */
/**
* Power function.
*
* @param x Base of the power.
* @param y Exponent of the power.
* @return x^y.
* The estimation of the relative error in the result is |log(x)*err(y)|+|y*err(x)/x|
*/
static public BigDecimal pow(final BigDecimal x, final BigDecimal y) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
throw new ArithmeticException("Cannot power negative " + x.toString());
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else {
/* return x^y = exp(y*log(x)) ;
*/
BigDecimal logx = log(x);
BigDecimal ylogx = y.multiply(logx);
BigDecimal resul = exp(ylogx);
/* The estimation of the relative error in the result is |log(x)*err(y)|+|y*err(x)/x|
*/
double errR = Math.abs(logx.doubleValue() * y.ulp().doubleValue() / 2.)
+ Math.abs(y.doubleValue() * x.ulp().doubleValue() / 2. / x.doubleValue());
MathContext mcR = new MathContext(err2prec(1.0, errR));
return resul.round(mcR);
}
} /* BigDecimalMath.pow */
/**
* Raise to an integer power and round.
*
* @param x The base.
* @param n The exponent.
* @return x^n.
*/
static public BigDecimal powRound(final BigDecimal x, final int n) {
/* The relative error in the result is n times the relative error in the input.
* The estimation is slightly optimistic due to the integer rounding of the logarithm.
*/
MathContext mc = new MathContext(x.precision() - (int) Math.log10((double) (Math.abs(n))));
return x.pow(n, mc);
} /* BigDecimalMath.powRound */
/**
* Trigonometric sine.
*
* @param x The argument in radians.
* @return sin(x) in the range -1 to 1.
*/
static public BigDecimal sin(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return sin(x.negate()).negate();
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else {
/* reduce modulo 2pi
*/
BigDecimal res = mod2pi(x);
double errpi = 0.5 * Math.abs(x.ulp().doubleValue());
int val = 2 + err2prec(FastMath.PI, errpi);
MathContext mc = new MathContext(val);
BigDecimal p = pi(mc);
mc = new MathContext(x.precision());
if (res.compareTo(p) > 0) {
/* pi 0) {
/* pi/2 0) {
/* x>pi/4: sin(x) = cos(pi/2-x)
*/
return cos(subtractRound(p.divide(new BigDecimal("2")), res));
} else {
/* Simple Taylor expansion, sum_{i=1..infinity} (-1)^(..)res^(2i+1)/(2i+1)! */
BigDecimal resul = res;
/* x^i */
BigDecimal xpowi = res;
/* 2i+1 factorial */
BigInteger ifac = BigInteger.ONE;
/* The error in the result is set by the error in x itself.
*/
double xUlpDbl = res.ulp().doubleValue();
/* The error in the result is set by the error in x itself.
* We need at most k terms to squeeze x^(2k+1)/(2k+1)! below this value.
* x^(2k+1) < x.ulp; (2k+1)*log10(x) < -x.precision; 2k*log10(x)< -x.precision;
* 2k*(-log10(x)) > x.precision; 2k*log10(1/x) > x.precision
*/
int k = (int) (res.precision() / Math.log10(1.0 / res.doubleValue())) / 2;
MathContext mcTay = new MathContext(err2prec(res.doubleValue(), xUlpDbl / k));
for (int i = 1; ; i++) {
/* TBD: at which precision will 2*i or 2*i+1 overflow?
*/
ifac = ifac.multiply(new BigInteger("" + (2 * i)));
ifac = ifac.multiply(new BigInteger("" + (2 * i + 1)));
xpowi = xpowi.multiply(res).multiply(res).negate();
BigDecimal corr = xpowi.divide(new BigDecimal(ifac), mcTay);
resul = resul.add(corr);
if (corr.abs().doubleValue() < 0.5 * xUlpDbl) {
break;
}
}
/* The error in the result is set by the error in x itself.
*/
mc = new MathContext(res.precision());
return resul.round(mc);
}
}
} /* sin */
}
/**
* Trigonometric cosine.
*
* @param x The argument in radians.
* @return cos(x) in the range -1 to 1.
*/
static public BigDecimal cos(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return cos(x.negate());
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ONE;
} else {
/* reduce modulo 2pi
*/
BigDecimal res = mod2pi(x);
double errpi = 0.5 * Math.abs(x.ulp().doubleValue());
int val = + err2prec(FastMath.PI, errpi);
MathContext mc = new MathContext(val);
BigDecimal p = pi(mc);
mc = new MathContext(x.precision());
if (res.compareTo(p) > 0) {
/* pi 0) {
/* pi/2 0) {
/* x>pi/4: cos(x) = sin(pi/2-x)
*/
return sin(subtractRound(p.divide(new BigDecimal("2")), res));
} else {
/* Simple Taylor expansion, sum_{i=0..infinity} (-1)^(..)res^(2i)/(2i)! */
BigDecimal resul = BigDecimal.ONE;
/* x^i */
BigDecimal xpowi = BigDecimal.ONE;
/* 2i factorial */
BigInteger ifac = BigInteger.ONE;
/* The absolute error in the result is the error in x^2/2 which is x times the error in x.
*/
double xUlpDbl = 0.5 * res.ulp().doubleValue() * res.doubleValue();
/* The error in the result is set by the error in x^2/2 itself, xUlpDbl.
* We need at most k terms to push x^(2k+1)/(2k+1)! below this value.
* x^(2k) < xUlpDbl; (2k)*log(x) < log(xUlpDbl);
*/
int k = (int) (Math.log(xUlpDbl) / Math.log(res.doubleValue())) / 2;
MathContext mcTay = new MathContext(err2prec(1., xUlpDbl / k));
for (int i = 1; ; i++) {
/* TBD: at which precision will 2*i-1 or 2*i overflow?
*/
ifac = ifac.multiply(new BigInteger("" + (2 * i - 1)));
ifac = ifac.multiply(new BigInteger("" + (2 * i)));
xpowi = xpowi.multiply(res).multiply(res).negate();
BigDecimal corr = xpowi.divide(new BigDecimal(ifac), mcTay);
resul = resul.add(corr);
if (corr.abs().doubleValue() < 0.5 * xUlpDbl) {
break;
}
}
/* The error in the result is governed by the error in x itself.
*/
mc = new MathContext(err2prec(resul.doubleValue(), xUlpDbl));
return resul.round(mc);
}
}
}
} /* BigDecimalMath.cos */
/**
* The trigonometric tangent.
*
* @param x the argument in radians.
* @return the tan(x)
*/
static public BigDecimal tan(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else if (x.compareTo(BigDecimal.ZERO) < 0) {
return tan(x.negate()).negate();
} else {
/* reduce modulo pi
*/
BigDecimal res = modpi(x);
/* absolute error in the result is err(x)/cos^2(x) to lowest order
*/
final double xDbl = res.doubleValue();
final double xUlpDbl = x.ulp().doubleValue() / 2.;
final double eps = xUlpDbl / 2. / Math.pow(Math.cos(xDbl), 2.);
if (xDbl > 0.8) {
/* tan(x) = 1/cot(x) */
BigDecimal co = cot(x);
MathContext mc = new MathContext(err2prec(1. / co.doubleValue(), eps));
return BigDecimal.ONE.divide(co, mc);
} else {
final BigDecimal xhighpr = scalePrec(res, 2);
final BigDecimal xhighprSq = multiplyRound(xhighpr, xhighpr);
BigDecimal resul = xhighpr.plus();
/* x^(2i+1) */
BigDecimal xpowi = xhighpr;
Bernoulli b = new Bernoulli();
/* 2^(2i) */
BigInteger fourn = new BigInteger("4");
/* (2i)! */
BigInteger fac = new BigInteger("2");
for (int i = 2; ; i++) {
Rational f = b.at(2 * i).abs();
fourn = fourn.shiftLeft(2);
fac = fac.multiply(new BigInteger("" + (2 * i))).multiply(new BigInteger("" + (2 * i - 1)));
f = f.multiply(fourn).multiply(fourn.subtract(BigInteger.ONE)).divide(fac);
xpowi = multiplyRound(xpowi, xhighprSq);
BigDecimal c = multiplyRound(xpowi, f);
resul = resul.add(c);
if (Math.abs(c.doubleValue()) < 0.1 * eps) {
break;
}
}
MathContext mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
}
}
} /* BigDecimalMath.tan */
/**
* The trigonometric co-tangent.
*
* @param x the argument in radians.
* @return the cot(x)
*/
static public BigDecimal cot(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) == 0) {
throw new ArithmeticException("Cannot take cot of zero " + x.toString());
} else if (x.compareTo(BigDecimal.ZERO) < 0) {
return cot(x.negate()).negate();
} else {
/* reduce modulo pi
*/
BigDecimal res = modpi(x);
/* absolute error in the result is err(x)/sin^2(x) to lowest order
*/
final double xDbl = res.doubleValue();
final double xUlpDbl = x.ulp().doubleValue() / 2.;
final double eps = xUlpDbl / 2. / Math.pow(Math.sin(xDbl), 2.);
final BigDecimal xhighpr = scalePrec(res, 2);
final BigDecimal xhighprSq = multiplyRound(xhighpr, xhighpr);
MathContext mc = new MathContext(err2prec(xhighpr.doubleValue(), eps));
BigDecimal resul = BigDecimal.ONE.divide(xhighpr, mc);
/* x^(2i-1) */
BigDecimal xpowi = xhighpr;
Bernoulli b = new Bernoulli();
/* 2^(2i) */
BigInteger fourn = new BigInteger("4");
/* (2i)! */
BigInteger fac = BigInteger.ONE;
for (int i = 1; ; i++) {
Rational f = b.at(2 * i);
fac = fac.multiply(new BigInteger("" + (2 * i))).multiply(new BigInteger("" + (2 * i - 1)));
f = f.multiply(fourn).divide(fac);
BigDecimal c = multiplyRound(xpowi, f);
if (i % 2 == 0) {
resul = resul.add(c);
} else {
resul = resul.subtract(c);
}
if (Math.abs(c.doubleValue()) < 0.1 * eps) {
break;
}
fourn = fourn.shiftLeft(2);
xpowi = multiplyRound(xpowi, xhighprSq);
}
mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
}
} /* BigDecimalMath.cot */
/**
* The inverse trigonometric sine.
*
* @param x the argument.
* @return the arcsin(x) in radians.
*/
static public BigDecimal asin(final BigDecimal x) {
if (x.compareTo(BigDecimal.ONE) > 0 || x.compareTo(BigDecimal.ONE.negate()) < 0) {
throw new ArithmeticException("Out of range argument " + x.toString() + " of asin");
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else if (x.compareTo(BigDecimal.ONE) == 0) {
/* arcsin(1) = pi/2
*/
double errpi = Math.sqrt(x.ulp().doubleValue());
MathContext mc = new MathContext(err2prec(3.14159, errpi));
return pi(mc).divide(new BigDecimal(2));
} else if (x.compareTo(BigDecimal.ZERO) < 0) {
return asin(x.negate()).negate();
} else if (x.doubleValue() > 0.7) {
final BigDecimal xCompl = BigDecimal.ONE.subtract(x);
final double xDbl = x.doubleValue();
final double xUlpDbl = x.ulp().doubleValue() / 2.;
final double eps = xUlpDbl / 2. / Math.sqrt(1. - Math.pow(xDbl, 2.));
final BigDecimal xhighpr = scalePrec(xCompl, 3);
final BigDecimal xhighprV = divideRound(xhighpr, 4);
BigDecimal resul = BigDecimal.ONE;
/* x^(2i+1) */
BigDecimal xpowi = BigDecimal.ONE;
/* i factorial */
BigInteger ifacN = BigInteger.ONE;
BigInteger ifacD = BigInteger.ONE;
for (int i = 1; ; i++) {
ifacN = ifacN.multiply(new BigInteger("" + (2 * i - 1)));
ifacD = ifacD.multiply(new BigInteger("" + i));
if (i == 1) {
xpowi = xhighprV;
} else {
xpowi = multiplyRound(xpowi, xhighprV);
}
BigDecimal c = divideRound(multiplyRound(xpowi, ifacN),
ifacD.multiply(new BigInteger("" + (2 * i + 1))));
resul = resul.add(c);
/* series started 1+x/12+... which yields an estimate of the sum’s error
*/
if (Math.abs(c.doubleValue()) < xUlpDbl / 120.) {
break;
}
}
/* sqrt(2*z)*(1+...)
*/
xpowi = sqrt(xhighpr.multiply(new BigDecimal(2)));
resul = multiplyRound(xpowi, resul);
MathContext mc = new MathContext(resul.precision());
BigDecimal pihalf = pi(mc).divide(new BigDecimal(2));
mc = new MathContext(err2prec(resul.doubleValue(), eps));
return pihalf.subtract(resul, mc);
} else {
/* absolute error in the result is err(x)/sqrt(1-x^2) to lowest order
*/
final double xDbl = x.doubleValue();
final double xUlpDbl = x.ulp().doubleValue() / 2.;
final double eps = xUlpDbl / 2. / Math.sqrt(1. - Math.pow(xDbl, 2.));
final BigDecimal xhighpr = scalePrec(x, 2);
final BigDecimal xhighprSq = multiplyRound(xhighpr, xhighpr);
BigDecimal resul = xhighpr.plus();
/* x^(2i+1) */
BigDecimal xpowi = xhighpr;
/* i factorial */
BigInteger ifacN = BigInteger.ONE;
BigInteger ifacD = BigInteger.ONE;
for (int i = 1; ; i++) {
ifacN = ifacN.multiply(new BigInteger("" + (2 * i - 1)));
ifacD = ifacD.multiply(new BigInteger("" + (2 * i)));
xpowi = multiplyRound(xpowi, xhighprSq);
BigDecimal c = divideRound(multiplyRound(xpowi, ifacN),
ifacD.multiply(new BigInteger("" + (2 * i + 1))));
resul = resul.add(c);
if (Math.abs(c.doubleValue()) < 0.1 * eps) {
break;
}
}
MathContext mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
}
} /* BigDecimalMath.asin */
/**
* The inverse trigonometric tangent.
*
* @param x the argument.
* @return the principal value of arctan(x) in radians in the range -pi/2 to +pi/2.
*/
static public BigDecimal atan(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return atan(x.negate()).negate();
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else if (x.doubleValue() > 0.7 && x.doubleValue() < 3.0) {
/* Abramowitz-Stegun 4.4.34 convergence acceleration
* 2*arctan(x) = arctan(2x/(1-x^2)) = arctan(y). x=(sqrt(1+y^2)-1)/y
* This maps 0<=y<=3 to 0<=x<=0.73 roughly. Temporarily with 2 protectionist digits.
*/
BigDecimal y = scalePrec(x, 2);
BigDecimal newx = divideRound(hypot(1, y).subtract(BigDecimal.ONE), y);
/* intermediate result with too optimistic error estimate*/
BigDecimal resul = multiplyRound(atan(newx), 2);
/* absolute error in the result is errx/(1+x^2), where errx = half of the ulp. */
double eps = x.ulp().doubleValue() / (2.0 * Math.hypot(1.0, x.doubleValue()));
MathContext mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
} else if (x.doubleValue() < 0.71) {
/* Taylor expansion around x=0; Abramowitz-Stegun 4.4.42 */
final BigDecimal xhighpr = scalePrec(x, 2);
final BigDecimal xhighprSq = multiplyRound(xhighpr, xhighpr).negate();
BigDecimal resul = xhighpr.plus();
/* signed x^(2i+1) */
BigDecimal xpowi = xhighpr;
/* absolute error in the result is errx/(1+x^2), where errx = half of the ulp.
*/
double eps = x.ulp().doubleValue() / (2.0 * Math.hypot(1.0, x.doubleValue()));
for (int i = 1; ; i++) {
xpowi = multiplyRound(xpowi, xhighprSq);
BigDecimal c = divideRound(xpowi, 2 * i + 1);
resul = resul.add(c);
if (Math.abs(c.doubleValue()) < 0.1 * eps) {
break;
}
}
MathContext mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
} else {
/* Taylor expansion around x=infinity; Abramowitz-Stegun 4.4.42 */
/* absolute error in the result is errx/(1+x^2), where errx = half of the ulp.
*/
double eps = x.ulp().doubleValue() / (2.0 * Math.hypot(1.0, x.doubleValue()));
/* start with the term pi/2; gather its precision relative to the expected result
*/
MathContext mc = new MathContext(2 + err2prec(3.1416, eps));
BigDecimal onepi = pi(mc);
BigDecimal resul = onepi.divide(new BigDecimal(2));
final BigDecimal xhighpr = divideRound(-1, scalePrec(x, 2));
final BigDecimal xhighprSq = multiplyRound(xhighpr, xhighpr).negate();
/* signed x^(2i+1) */
BigDecimal xpowi = xhighpr;
for (int i = 0; ; i++) {
BigDecimal c = divideRound(xpowi, 2 * i + 1);
resul = resul.add(c);
if (Math.abs(c.doubleValue()) < 0.1 * eps) {
break;
}
xpowi = multiplyRound(xpowi, xhighprSq);
}
mc = new MathContext(err2prec(resul.doubleValue(), eps));
return resul.round(mc);
}
} /* BigDecimalMath.atan */
/**
* The hyperbolic cosine.
*
* @param x The argument.
* @return The cosh(x) = (exp(x)+exp(-x))/2 .
*/
static public BigDecimal cosh(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return cos(x.negate());
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ONE;
} else {
if (x.doubleValue() > 1.5) {
/* cosh^2(x) = 1+ sinh^2(x).
*/
return hypot(1, sinh(x));
} else {
BigDecimal xhighpr = scalePrec(x, 2);
/* Simple Taylor expansion, sum_{0=1..infinity} x^(2i)/(2i)! */
BigDecimal resul = BigDecimal.ONE;
/* x^i */
BigDecimal xpowi = BigDecimal.ONE;
/* 2i factorial */
BigInteger ifac = BigInteger.ONE;
/* The absolute error in the result is the error in x^2/2 which is x times the error in x.
*/
double xUlpDbl = 0.5 * x.ulp().doubleValue() * x.doubleValue();
/* The error in the result is set by the error in x^2/2 itself, xUlpDbl.
* We need at most k terms to push x^(2k)/(2k)! below this value.
* x^(2k) < xUlpDbl; (2k)*log(x) < log(xUlpDbl);
*/
int k = (int) (Math.log(xUlpDbl) / Math.log(x.doubleValue())) / 2;
/* The individual terms are all smaller than 1, so an estimate of 1.0 for
* the absolute value will give a safe relative error estimate for the indivdual terms
*/
MathContext mcTay = new MathContext(err2prec(1., xUlpDbl / k));
for (int i = 1; ; i++) {
/* TBD: at which precision will 2*i-1 or 2*i overflow?
*/
ifac = ifac.multiply(new BigInteger("" + (2 * i - 1)));
ifac = ifac.multiply(new BigInteger("" + (2 * i)));
xpowi = xpowi.multiply(xhighpr).multiply(xhighpr);
BigDecimal corr = xpowi.divide(new BigDecimal(ifac), mcTay);
resul = resul.add(corr);
if (corr.abs().doubleValue() < 0.5 * xUlpDbl) {
break;
}
} /* The error in the result is governed by the error in x itself.
*/
MathContext mc = new MathContext(err2prec(resul.doubleValue(), xUlpDbl));
return resul.round(mc);
}
}
} /* BigDecimalMath.cosh */
/**
* The hyperbolic sine.
*
* @param x the argument.
* @return the sinh(x) = (exp(x)-exp(-x))/2 .
*/
static public BigDecimal sinh(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return sinh(x.negate()).negate();
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else {
if (x.doubleValue() > 2.4) {
/* Move closer to zero with sinh(2x)= 2*sinh(x)*cosh(x).
*/
BigDecimal two = new BigDecimal(2);
BigDecimal xhalf = x.divide(two);
BigDecimal resul = sinh(xhalf).multiply(cosh(xhalf)).multiply(two);
/* The error in the result is set by the error in x itself.
* The first derivative of sinh(x) is cosh(x), so the absolute error
* in the result is cosh(x)*errx, and the relative error is coth(x)*errx = errx/tanh(x)
*/
double eps = Math.tanh(x.doubleValue());
MathContext mc = new MathContext(err2prec(0.5 * x.ulp().doubleValue() / eps));
return resul.round(mc);
} else {
BigDecimal xhighpr = scalePrec(x, 2);
/* Simple Taylor expansion, sum_{i=0..infinity} x^(2i+1)/(2i+1)! */
BigDecimal resul = xhighpr;
/* x^i */
BigDecimal xpowi = xhighpr;
/* 2i+1 factorial */
BigInteger ifac = BigInteger.ONE;
/* The error in the result is set by the error in x itself.
*/
double xUlpDbl = x.ulp().doubleValue();
/* The error in the result is set by the error in x itself.
* We need at most k terms to squeeze x^(2k+1)/(2k+1)! below this value.
* x^(2k+1) < x.ulp; (2k+1)*log10(x) < -x.precision; 2k*log10(x)< -x.precision;
* 2k*(-log10(x)) > x.precision; 2k*log10(1/x) > x.precision
*/
int k = (int) (x.precision() / Math.log10(1.0 / xhighpr.doubleValue())) / 2;
MathContext mcTay = new MathContext(err2prec(x.doubleValue(), xUlpDbl / k));
for (int i = 1; ; i++) {
/* TBD: at which precision will 2*i or 2*i+1 overflow?
*/
ifac = ifac.multiply(new BigInteger("" + (2 * i)));
ifac = ifac.multiply(new BigInteger("" + (2 * i + 1)));
xpowi = xpowi.multiply(xhighpr).multiply(xhighpr);
BigDecimal corr = xpowi.divide(new BigDecimal(ifac), mcTay);
resul = resul.add(corr);
if (corr.abs().doubleValue() < 0.5 * xUlpDbl) {
break;
}
} /* The error in the result is set by the error in x itself.
*/
MathContext mc = new MathContext(x.precision());
return resul.round(mc);
}
}
} /* BigDecimalMath.sinh */
/**
* The hyperbolic tangent.
*
* @param x The argument.
* @return The tanh(x) = sinh(x)/cosh(x).
*/
static public BigDecimal tanh(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) < 0) {
return tanh(x.negate()).negate();
} else if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else {
BigDecimal xhighpr = scalePrec(x, 2);
/* tanh(x) = (1-e^(-2x))/(1+e^(-2x)) .
*/
BigDecimal exp2x = exp(xhighpr.multiply(new BigDecimal(-2)));
/* The error in tanh x is err(x)/cosh^2(x).
*/
double eps = 0.5 * x.ulp().doubleValue() / Math.pow(Math.cosh(x.doubleValue()), 2.0);
MathContext mc = new MathContext(err2prec(Math.tanh(x.doubleValue()), eps));
return BigDecimal.ONE.subtract(exp2x).divide(BigDecimal.ONE.add(exp2x), mc);
}
} /* BigDecimalMath.tanh */
/**
* The inverse hyperbolic sine.
*
* @param x The argument.
* @return The arcsinh(x) .
*/
static public BigDecimal asinh(final BigDecimal x) {
if (x.compareTo(BigDecimal.ZERO) == 0) {
return BigDecimal.ZERO;
} else {
BigDecimal xhighpr = scalePrec(x, 2);
/* arcsinh(x) = log(x+hypot(1,x))
*/
BigDecimal logx = log(hypot(1, xhighpr).add(xhighpr));
/* The absolute error in arcsinh x is err(x)/sqrt(1+x^2)
*/
double xDbl = x.doubleValue();
double eps = 0.5 * x.ulp().doubleValue() / Math.hypot(1., xDbl);
MathContext mc = new MathContext(err2prec(logx.doubleValue(), eps));
return logx.round(mc);
}
} /* BigDecimalMath.asinh */
/**
* The inverse hyperbolic cosine.
*
* @param x The argument.
* @return The arccosh(x) .
*/
static public BigDecimal acosh(final BigDecimal x) {
if (x.compareTo(BigDecimal.ONE) < 0) {
throw new ArithmeticException("Out of range argument cosh " + x.toString());
} else if (x.compareTo(BigDecimal.ONE) == 0) {
return BigDecimal.ZERO;
} else {
BigDecimal xhighpr = scalePrec(x, 2);
/* arccosh(x) = log(x+sqrt(x^2-1))
*/
BigDecimal logx = log(sqrt(xhighpr.pow(2).subtract(BigDecimal.ONE)).add(xhighpr));
/* The absolute error in arcsinh x is err(x)/sqrt(x^2-1)
*/
double xDbl = x.doubleValue();
double eps = 0.5 * x.ulp().doubleValue() / Math.sqrt(xDbl * xDbl - 1.);
MathContext mc = new MathContext(err2prec(logx.doubleValue(), eps));
return logx.round(mc);
}
} /* BigDecimalMath.acosh */
/**
* The Gamma function.
*
* @param x The argument.
* @return Gamma(x).
*/
static public BigDecimal Gamma(final BigDecimal x) {
/* reduce to interval near 1.0 with the functional relation, Abramowitz-Stegun 6.1.33
*/
if (x.compareTo(BigDecimal.ZERO) < 0) {
return divideRound(Gamma(x.add(BigDecimal.ONE)), x);
} else if (x.doubleValue() > 1.5) {
/* Gamma(x) = Gamma(xmin+n) = Gamma(xmin)*Pochhammer(xmin,n).
*/
int n = (int) (x.doubleValue() - 0.5);
BigDecimal xmin1 = x.subtract(new BigDecimal(n));
return multiplyRound(Gamma(xmin1), pochhammer(xmin1, n));
} else {
/* apply Abramowitz-Stegun 6.1.33
*/
BigDecimal z = x.subtract(BigDecimal.ONE);
/* add intermediately 2 digits to the partial sum accumulation
*/
z = scalePrec(z, 2);
MathContext mcloc = new MathContext(z.precision());
/* measure of the absolute error is the relative error in the first, logarithmic term
*/
double eps = x.ulp().doubleValue() / x.doubleValue();
BigDecimal resul = log(scalePrec(x, 2)).negate();
if (x.compareTo(BigDecimal.ONE) != 0) {
BigDecimal gammCompl = BigDecimal.ONE.subtract(gamma(mcloc));
resul = resul.add(multiplyRound(z, gammCompl));
for (int n = 2; ; n++) {
/* multiplying z^n/n by zeta(n-1) means that the two relative errors add.
* so the requirement in the relative error of zeta(n)-1 is that this is somewhat
* smaller than the relative error in z^n/n (the absolute error of thelatter is the
* absolute error in z)
*/
BigDecimal c = divideRound(z.pow(n, mcloc), n);
MathContext m = new MathContext(err2prec(n * z.ulp().doubleValue() / 2. / z.doubleValue()));
c = c.round(m);
/* At larger n, zeta(n)-1 is roughly 1/2^n. The product is c/2^n.
* The relative error in c is c.ulp/2/c . The error in the product should be small versus eps/10.
* Error from 1/2^n is c*err(sigma-1).
* We need a relative error of zeta-1 of the order of c.ulp/50/c. This is an absolute
* error in zeta-1 of c.ulp/50/c/2^n, and also the absolute error in zeta, because zeta is
* of the order of 1.
*/
if (eps / 100. / c.doubleValue() < 0.01) {
m = new MathContext(err2prec(eps / 100. / c.doubleValue()));
} else {
m = new MathContext(2);
}
/* zeta(n) -1 */
BigDecimal zetm1 = zeta(n, m).subtract(BigDecimal.ONE);
c = multiplyRound(c, zetm1);
if (n % 2 == 0) {
resul = resul.add(c);
} else {
resul = resul.subtract(c);
}
/* alternating sum, so truncating as eps is reached suffices
*/
if (Math.abs(c.doubleValue()) < eps) {
break;
}
}
}
/* The relative error in the result is the absolute error in the
* input variable times the digamma (psi) value at that point.
*/
double psi = 0.5772156649;
double zdbl = z.doubleValue();
for (int n = 1; n
< 5; n++) {
psi += zdbl / n / (n + zdbl);
}
eps = psi * x.ulp().doubleValue() / 2.;
mcloc = new MathContext(err2prec(eps));
return exp(resul).round(mcloc);
}
} /* BigDecimalMath.gamma */
/**
* Pochhammer’s function.
*
* @param x The main argument.
* @param n The non-negative index.
* @return (x)_n = x(x+1)(x+2)*...*(x+n-1).
*/
static public BigDecimal pochhammer(final BigDecimal x, final int n) {
/* reduce to interval near 1.0 with the functional relation, Abramowitz-Stegun 6.1.33
*/
if (n < 0) {
throw new ProviderException("Unimplemented pochhammer with negative index " + n);
} else if (n == 0) {
return BigDecimal.ONE;
} else {
/* internally two safety digits
*/
BigDecimal xhighpr = scalePrec(x, 2);
BigDecimal resul = xhighpr;
double xUlpDbl = x.ulp().doubleValue();
double xDbl = x.doubleValue();
/* relative error of the result is the sum of the relative errors of the factors
*/
double eps = 0.5 * xUlpDbl / Math.abs(xDbl);
for (int i = 1; i
< n; i++) {
eps += 0.5 * xUlpDbl / Math.abs(xDbl + i);
resul = resul.multiply(xhighpr.add(new BigDecimal(i)));
final MathContext mcloc = new MathContext(4 + err2prec(eps));
resul = resul.round(mcloc);
}
return resul.round(new MathContext(err2prec(eps)));
}
} /* BigDecimalMath.pochhammer */
/**
* Reduce value to the interval [0,2*Pi].
*
* @param x the original value
* @return the value modulo 2*pi in the interval from 0 to 2*pi.
*/
static public BigDecimal mod2pi(BigDecimal x) {
/* write x= 2*pi*k+r with the precision in r defined by the precision of x and not
* compromised by the precision of 2*pi, so the ulp of 2*pi*k should match the ulp of x.
* First getFloat a guess of k to figure out how many digits of 2*pi are needed.
*/
int k = (int) (0.5 * x.doubleValue() / Math.PI);
/* want to have err(2*pi*k)< err(x)=0.5*x.ulp, so err(pi) = err(x)/(4k) with two safety digits
*/
double err2pi;
if (k != 0) {
err2pi = 0.25 * Math.abs(x.ulp().doubleValue() / k);
} else {
err2pi = 0.5 * Math.abs(x.ulp().doubleValue());
}
MathContext mc = new MathContext(2 + err2prec(6.283, err2pi));
BigDecimal twopi = pi(mc).multiply(new BigDecimal(2));
/* Delegate the actual operation to the BigDecimal class, which may return
* a negative value of x was negative .
*/
BigDecimal res = x.remainder(twopi);
if (res.compareTo(BigDecimal.ZERO) < 0) {
res = res.add(twopi);
}
/* The actual precision is set by the input value, its absolute value of x.ulp()/2.
*/
mc = new MathContext(err2prec(res.doubleValue(), x.ulp().doubleValue() / 2.));
return res.round(mc);
} /* mod2pi */
/**
* Reduce value to the interval [-Pi/2,Pi/2].
*
* @param x The original value
* @return The value modulo pi, shifted to the interval from -Pi/2 to Pi/2.
*/
static public BigDecimal modpi(BigDecimal x) {
/* write x= pi*k+r with the precision in r defined by the precision of x and not
* compromised by the precision of pi, so the ulp of pi*k should match the ulp of x.
* First getFloat a guess of k to figure out how many digits of pi are needed.
*/
int k = (int) (x.doubleValue() / Math.PI);
/* want to have err(pi*k)< err(x)=x.ulp/2, so err(pi) = err(x)/(2k) with two safety digits
*/
double errpi;
if (k != 0) {
errpi = 0.5 * Math.abs(x.ulp().doubleValue() / k);
} else {
errpi = 0.5 * Math.abs(x.ulp().doubleValue());
}
MathContext mc = new MathContext(2 + err2prec(3.1416, errpi));
BigDecimal onepi = pi(mc);
BigDecimal pihalf = onepi.divide(new BigDecimal(2));
/* Delegate the actual operation to the BigDecimal class, which may return
* a negative value of x was negative .
*/
BigDecimal res = x.remainder(onepi);
if (res.compareTo(pihalf) > 0) {
res = res.subtract(onepi);
} else if (res.compareTo(pihalf.negate()) < 0) {
res = res.add(onepi);
}
/* The actual precision is set by the input value, its absolute value of x.ulp()/2.
*/
mc = new MathContext(err2prec(res.doubleValue(), x.ulp().doubleValue() / 2.));
return res.round(mc);
} /* modpi */
/**
* Riemann zeta function.
*
* @param n The positive integer argument.
* 32
* @param mc Specification of the accuracy of the result.
* @return zeta(n).
*/
static public BigDecimal zeta(final int n, final MathContext mc) {
if (n <= 0) {
throw new ProviderException("Unimplemented zeta at negative argument " + n);
}
if (n == 1) {
throw new ArithmeticException("Pole at zeta(1) ");
}
if (n % 2 == 0) {
/* Even indices. Abramowitz-Stegun 23.2.16. Start with 2^(n-1)*B(n)/n!
*/
Rational b = (new Bernoulli()).at(n).abs();
b = b.divide((new Factorial()).at(n));
b = b.multiply(BigInteger.ONE.shiftLeft(n - 1));
/* to be multiplied by pi^n. Absolute error in the result of pi^n is n times
* error in pi times pi^(n-1). Relative error is n*error(pi)/pi, requested by mc.
* Need one more digit in pi if n=10, two digits if n=100 etc, and add one extra digit.
*/
MathContext mcpi = new MathContext(mc.getPrecision() + (int) (Math.log10(10.0 * n)));
final BigDecimal piton = pi(mcpi).pow(n, mc);
return multiplyRound(piton, b);
} else if (n == 3) {
/* Broadhurst BBP \protect\vrule width0pt\protect\href{http://arxiv.org/abs/math/9803067}{arXiv:math/9803067}
* Error propagation: S31 is roughly 0.087, S33 roughly 0.131
*/
int[] a31 = {1, -7, -1, 10, -1, -7, 1, 0};
int[] a33 = {1, 1, -1, -2, -1, 1, 1, 0};
BigDecimal S31 = broadhurstBBP(3, 1, a31, mc);
BigDecimal S33 = broadhurstBBP(3, 3, a33, mc);
S31 = S31.multiply(new BigDecimal(48));
S33 = S33.multiply(new BigDecimal(32));
return S31.add(S33).divide(new BigDecimal(7), mc);
} else if (n == 5) {
/* Broadhurst BBP \protect\vrule width0pt\protect\href{http://arxiv.org/abs/math/9803067}{arXiv:math/9803067}
* Error propagation: S51 is roughly -11.15, S53 roughly 22.165, S55 is roughly 0.031
* 9*2048*S51/6265 = -3.28. 7*2038*S53/61651= 5.07. 738*2048*S55/61651= 0.747.
* The result is of the order 1.03, so we add 2 digits to S51 and S52 and one digit to S55.
*/
int[] a51 = {31, -1614, -31, -6212, -31, -1614, 31, 74552};
int[] a53 = {173, 284, -173, -457, -173, 284, 173, -111};
int[] a55 = {1, 0, -1, -1, -1, 0, 1, 1};
BigDecimal S51 = broadhurstBBP(5, 1, a51, new MathContext(2 + mc.getPrecision()));
BigDecimal S53 = broadhurstBBP(5, 3, a53, new MathContext(2 + mc.getPrecision()));
BigDecimal S55 = broadhurstBBP(5, 5, a55, new MathContext(1 + mc.getPrecision()));
S51 = S51.multiply(new BigDecimal(18432));
S53 = S53.multiply(new BigDecimal(14336));
S55 = S55.multiply(new BigDecimal(1511424));
return S51.add(S53).subtract(S55).divide(new BigDecimal(62651), mc);
} else {
/* Cohen et al Exp Math 1 (1) (1992) 25
*/
Rational betsum = new Rational();
Bernoulli bern = new Bernoulli();
Factorial fact = new Factorial();
for (int npr = 0; npr
<= (n + 1) / 2; npr++) {
Rational b = bern.at(2 * npr).multiply(bern.at(n + 1 - 2 * npr));
b = b.divide(fact.at(2 * npr)).divide(fact.at(n + 1 - 2 * npr));
b = b.multiply(1 - 2 * npr);
if (npr % 2 == 0) {
betsum = betsum.add(b);
} else {
betsum = betsum.subtract(b);
}
}
betsum = betsum.divide(n - 1);
/* The first term, including the facor (2pi)^n, is essentially most
* of the result, near one. The second term below is roughly in the range 0.003 to 0.009.
* So the precision here is matching the precisionn requested by mc, and the precision
* requested for 2*pi is in absolute terms adjusted.
*/
MathContext mcloc = new MathContext(2 + mc.getPrecision() + (int) (Math.log10((double) (n))));
BigDecimal ftrm = pi(mcloc).multiply(new BigDecimal(2));
ftrm = ftrm.pow(n);
ftrm = multiplyRound(ftrm, betsum.BigDecimalValue(mcloc));
BigDecimal exps = new BigDecimal(0);
/* the basic accuracy of the accumulated terms before multiplication with 2
*/
double eps = Math.pow(10., -mc.getPrecision());
if (n % 4 == 3) {
/* since the argument n is at least 7 here, the drop
* of the terms is at rather constant pace at least 10^-3, for example
* 0.0018, 0.2e-7, 0.29e-11, 0.74e-15 etc for npr=1,2,3.... We want 2 times these terms
* fall below eps/10.
*/
int kmax = mc.getPrecision() / 3;
eps /= kmax;
/* need an error of eps for 2/(exp(2pi)-1) = 0.0037
* The absolute error is 4*exp(2pi)*err(pi)/(exp(2pi)-1)^2=0.0075*err(pi)
*/
BigDecimal exp2p = pi(new MathContext(3 + err2prec(3.14, eps / 0.0075)));
exp2p = exp(exp2p.multiply(new BigDecimal(2)));
BigDecimal c = exp2p.subtract(BigDecimal.ONE);
exps = divideRound(1, c);
for (int npr = 2; npr
<= kmax; npr++) {
/* the error estimate above for npr=1 is the worst case of
* the absolute error created by an error in 2pi. So we can
* safely re-use the exp2p value computed above without
* reassessment of its error.
*/
c = powRound(exp2p, npr).subtract(BigDecimal.ONE);
c = multiplyRound(c, (new BigInteger("" + npr)).pow(n));
c = divideRound(1, c);
exps = exps.add(c);
}
} else {
/* since the argument n is at least 9 here, the drop
* of the terms is at rather constant pace at least 10^-3, for example
* 0.0096, 0.5e-7, 0.3e-11, 0.6e-15 etc. We want these terms
* fall below eps/10.
*/
int kmax = (1 + mc.getPrecision()) / 3;
eps /= kmax;
/* need an error of eps for 2/(exp(2pi)-1)*(1+4*Pi/8/(1-exp(-2pi)) = 0.0096
* at k=7 or = 0.00766 at k=13 for example.
* The absolute error is 0.017*err(pi) at k=9, 0.013*err(pi) at k=13, 0.012 at k=17
*/
BigDecimal twop = pi(new MathContext(3 + err2prec(3.14, eps / 0.017)));
twop = twop.multiply(new BigDecimal(2));
BigDecimal exp2p = exp(twop);
BigDecimal c = exp2p.subtract(BigDecimal.ONE);
exps = divideRound(1, c);
c = BigDecimal.ONE.subtract(divideRound(1, exp2p));
c = divideRound(twop, c).multiply(new BigDecimal(2));
c = divideRound(c, n - 1).add(BigDecimal.ONE);
exps = multiplyRound(exps, c);
for (int npr = 2; npr
<= kmax; npr++) {
c = powRound(exp2p, npr).subtract(BigDecimal.ONE);
c = multiplyRound(c, (new BigInteger("" + npr)).pow(n));
BigDecimal d = divideRound(1, exp2p.pow(npr));
d = BigDecimal.ONE.subtract(d);
d = divideRound(twop, d).multiply(new BigDecimal(2 * npr));
d = divideRound(d, n - 1).add(BigDecimal.ONE);
d = divideRound(d, c);
exps = exps.add(d);
}
}
exps = exps.multiply(new BigDecimal(2));
return ftrm.subtract(exps, mc);
}
} /* zeta */
/**
* Riemann zeta function.
*
* @param n The positive integer argument.
* @return zeta(n)-1.
*/
static public double zeta1(final int n) {
/* precomputed static table in double precision
*/
final double[] zmin1 = {0., 0.,
6.449340668482264364724151666e-01,
2.020569031595942853997381615e-01, 8.232323371113819151600369654e-02,
3.692775514336992633136548646e-02, 1.734306198444913971451792979e-02,
8.349277381922826839797549850e-03, 4.077356197944339378685238509e-03,
2.008392826082214417852769232e-03, 9.945751278180853371459589003e-04,
4.941886041194645587022825265e-04, 2.460865533080482986379980477e-04,
1.227133475784891467518365264e-04, 6.124813505870482925854510514e-05,
3.058823630702049355172851064e-05, 1.528225940865187173257148764e-05,
7.637197637899762273600293563e-06, 3.817293264999839856461644622e-06,
1.908212716553938925656957795e-06, 9.539620338727961131520386834e-07,
4.769329867878064631167196044e-07, 2.384505027277329900036481868e-07,
1.192199259653110730677887189e-07, 5.960818905125947961244020794e-08,
2.980350351465228018606370507e-08, 1.490155482836504123465850663e-08,
7.450711789835429491981004171e-09, 3.725334024788457054819204018e-09,
1.862659723513049006403909945e-09, 9.313274324196681828717647350e-10,
4.656629065033784072989233251e-10, 2.328311833676505492001455976e-10,
1.164155017270051977592973835e-10, 5.820772087902700889243685989e-11,
2.910385044497099686929425228e-11, 1.455192189104198423592963225e-11,
7.275959835057481014520869012e-12, 3.637979547378651190237236356e-12,
1.818989650307065947584832101e-12, 9.094947840263889282533118387e-13,
4.547473783042154026799112029e-13, 2.273736845824652515226821578e-13,
1.136868407680227849349104838e-13, 5.684341987627585609277182968e-14,
2.842170976889301855455073705e-14, 1.421085482803160676983430714e-14,
7.105427395210852712877354480e-15, 3.552713691337113673298469534e-15,
1.776356843579120327473349014e-15, 8.881784210930815903096091386e-16,
4.440892103143813364197770940e-16, 2.220446050798041983999320094e-16,
1.110223025141066133720544570e-16, 5.551115124845481243723736590e-17,
2.775557562136124172581632454e-17, 1.387778780972523276283909491e-17,
6.938893904544153697446085326e-18, 3.469446952165922624744271496e-18,
1.734723476047576572048972970e-18, 8.673617380119933728342055067e-19,
4.336808690020650487497023566e-19, 2.168404344997219785013910168e-19,
1.084202172494241406301271117e-19, 5.421010862456645410918700404e-20,
2.710505431223468831954621312e-20, 1.355252715610116458148523400e-20,
6.776263578045189097995298742e-21, 3.388131789020796818085703100e-21,
1.694065894509799165406492747e-21, 8.470329472546998348246992609e-22,
4.235164736272833347862270483e-22, 2.117582368136194731844209440e-22,
1.058791184068023385226500154e-22, 5.293955920339870323813912303e-23,
2.646977960169852961134116684e-23, 1.323488980084899080309451025e-23,
6.617444900424404067355245332e-24, 3.308722450212171588946956384e-24,
1.654361225106075646229923677e-24, 8.271806125530344403671105617e-25,
4.135903062765160926009382456e-25, 2.067951531382576704395967919e-25,
1.033975765691287099328409559e-25, 5.169878828456431320410133217e-26,
2.584939414228214268127761771e-26, 1.292469707114106670038112612e-26,
6.462348535570531803438002161e-27, 3.231174267785265386134814118e-27,
1.615587133892632521206011406e-27, 8.077935669463162033158738186e-28,
4.038967834731580825622262813e-28, 2.019483917365790349158762647e-28,
1.009741958682895153361925070e-28, 5.048709793414475696084771173e-29,
2.524354896707237824467434194e-29, 1.262177448353618904375399966e-29,
6.310887241768094495682609390e-30, 3.155443620884047239109841220e-30,
1.577721810442023616644432780e-30, 7.888609052210118073520537800e-31
};
if (n <= 0) {
throw new ProviderException("Unimplemented zeta at negative argument " + n);
}
if (n == 1) {
throw new ArithmeticException("Pole at zeta(1) ");
}
if (n < zmin1.length) /* look it up if available */ {
return zmin1[n];
} else {
/* Result is roughly 2^(-n), desired accuracy 18 digits. If zeta(n) is computed, the equivalent accuracy
* in relative units is higher, because zeta is around 1.
*/
double eps = 1.e-18 * Math.pow(2., (double) (-n));
MathContext mc = new MathContext(err2prec(eps));
return zeta(n, mc).subtract(BigDecimal.ONE).doubleValue();
}
} /* zeta */
/**
* Broadhurst ladder sequence.
*
* @param a The vector of 8 integer arguments
* @param mc Specification of the accuracy of the result
* @return S_(n, p)(a)
* @see \protect\vrule width0pt\protect\href{http://arxiv.org/abs/math/9803067}{arXiv:math/9803067}
*/
static protected BigDecimal broadhurstBBP(final int n, final int p, final int a[], MathContext mc) {
/* Explore the actual magnitude of the result first with a quick estimate.
*/
double x = 0.0;
for (int k = 1; k
< 10; k++) {
x += a[(k - 1) % 8] / Math.pow(2., p * (k + 1) / 2) / Math.pow((double) k, n);
}
/* Convert the relative precision and estimate of the result into an absolute precision.
*/
double eps = prec2err(x, mc.getPrecision());
/* Divide this through the number of terms in the sum to account for error accumulation
* The divisor 2^(p(k+1)/2) means that on the average each 8th term in k has shrunk by
* relative to the 8th predecessor by 1/2^(4p). 1/2^(4pc) = 10^(-precision) with c the 8term
* cycles yields c=log_2( 10^precision)/4p = 3.3*precision/4p with k=8c
*/
int kmax = (int) (6.6 * mc.getPrecision() / p);
/* Now eps is the absolute error in each term */
eps /= kmax;
BigDecimal res = BigDecimal.ZERO;
for (int c = 0; ; c++) {
Rational r = new Rational();
for (int k = 0; k
< 8; k++) {
Rational tmp = new Rational(new BigInteger("" + a[k]), (new BigInteger("" + (1 + 8 * c + k))).pow(n));
/* floor( (pk+p)/2)
*/
int pk1h = p * (2 + 8 * c + k) / 2;
tmp = tmp.divide(BigInteger.ONE.shiftLeft(pk1h));
r = r.add(tmp);
}
if (Math.abs(r.doubleValue()) < eps) {
break;
}
MathContext mcloc = new MathContext(1 + err2prec(r.doubleValue(), eps));
res = res.add(r.BigDecimalValue(mcloc));
}
return res.round(mc);
} /* broadhurstBBP */
/**
* Add and round according to the larger of the two ulp’s.
*
* @param x The left summand
* @param y The right summand
* @return The sum x+y.
*/
static public BigDecimal addRound(final BigDecimal x, final BigDecimal y) {
BigDecimal resul = x.add(y);
/* The estimation of the absolute error in the result is |err(y)|+|err(x)|
*/
double errR = Math.abs(y.ulp().doubleValue() / 2.) + Math.abs(x.ulp().doubleValue() / 2.);
MathContext mc = new MathContext(err2prec(resul.doubleValue(), errR));
return resul.round(mc);
} /* addRound */
/**
* Subtract and round according to the larger of the two ulp’s.
*
* @param x The left term.
* @param y The right term.
* @return The difference x-y.
*/
static public BigDecimal subtractRound(final BigDecimal x, final BigDecimal y) {
BigDecimal resul = x.subtract(y);
/* The estimation of the absolute error in the result is |err(y)|+|err(x)|
*/
double errR = Math.abs(y.ulp().doubleValue() / 2.) + Math.abs(x.ulp().doubleValue() / 2.);
MathContext mc = new MathContext(err2prec(resul.doubleValue(), errR));
return resul.round(mc);
} /* subtractRound */
/**
* Multiply and round.
*
* @param x The left factor.
* @param y The right factor.
* @return The product x*y.
*/
static public BigDecimal multiplyRound(final BigDecimal x, final BigDecimal y) {
BigDecimal resul = x.multiply(y);
/* The estimation of the relative error in the result is the sum of the relative
* errors |err(y)/y|+|err(x)/x|
*/
MathContext mc = new MathContext(Math.min(x.precision(), y.precision()));
return resul.round(mc);
} /* multiplyRound */
/**
* Multiply and round.
*
* @param x The left factor.
* @param f The right factor.
* @return The product x*f.
*/
static public BigDecimal multiplyRound(final BigDecimal x, final Rational f) {
if (f.compareTo(BigInteger.ZERO) == 0) {
return BigDecimal.ZERO;
} else {
/* Convert the rational value with two digits of extra precision
*/
MathContext mc = new MathContext(2 + x.precision());
BigDecimal fbd = f.BigDecimalValue(mc);
/* and the precision of the product is then dominated by the precision in x
*/
return multiplyRound(x, fbd);
}
}
/**
* Multiply and round.
*
* @param x The left factor.
* @param n The right factor.
* @return The product x*n.
*/
static public BigDecimal multiplyRound(final BigDecimal x, final int n) {
BigDecimal resul = x.multiply(new BigDecimal(n));
/* The estimation of the absolute error in the result is |n*err(x)|
*/
MathContext mc = new MathContext(n != 0 ? x.precision() : 0);
return resul.round(mc);
}
/**
* Multiply and round.
*
* @param x The left factor.
* @param n The right factor.
* @return the product x*n
*/
static public BigDecimal multiplyRound(final BigDecimal x, final BigInteger n) {
BigDecimal resul = x.multiply(new BigDecimal(n));
/* The estimation of the absolute error in the result is |n*err(x)|
*/
MathContext mc = new MathContext(n.compareTo(BigInteger.ZERO) != 0 ? x.precision() : 0);
return resul.round(mc);
}
/**
* Divide and round.
*
* @param x The numerator
* @param y The denominator
* @return the divided x/y
*/
static public BigDecimal divideRound(final BigDecimal x, final BigDecimal y) {
/* The estimation of the relative error in the result is |err(y)/y|+|err(x)/x|
*/
MathContext mc = new MathContext(Math.min(x.precision(), y.precision()));
return x.divide(y, mc);
}
/**
* Divide and round.
*
* @param x The numerator
* @param n The denominator
* @return the divided x/n
*/
static public BigDecimal divideRound(final BigDecimal x, final int n) {
/* The estimation of the relative error in the result is |err(x)/x|
*/
MathContext mc = new MathContext(x.precision());
return x.divide(new BigDecimal(n), mc);
}
/**
* Divide and round.
*
* @param x The numerator
* @param n The denominator
* @return the divided x/n
*/
static public BigDecimal divideRound(final BigDecimal x, final BigInteger n) {
/* The estimation of the relative error in the result is |err(x)/x|
*/
MathContext mc = new MathContext(x.precision());
return x.divide(new BigDecimal(n), mc);
}
/**
* Divide and round.
*
* @param n The numerator
* @param x The denominator
* @return the divided n/x
*/
static public BigDecimal divideRound(final BigInteger n, final BigDecimal x) {
/* The estimation of the relative error in the result is |err(x)/x|
*/
MathContext mc = new MathContext(x.precision());
return new BigDecimal(n).divide(x, mc);
}
/**
* Divide and round.
*
* @param n The numerator.
* @param x The denominator.
* @return the divided n/x.
*/
static public BigDecimal divideRound(final int n, final BigDecimal x) {
/* The estimation of the relative error in the result is |err(x)/x|
*/
MathContext mc = new MathContext(x.precision());
return new BigDecimal(n).divide(x, mc);
}
/**
* Append decimal zeros to the value. This returns a value which appears to have
* a higher precision than the input.
*
* @param x The input value
* @param d The (positive) value of zeros to be added as least significant digits.
* @return The same value as the input but with increased (pseudo) precision.
*/
static public BigDecimal scalePrec(final BigDecimal x, int d) {
return x.setScale(d + x.scale());
}
/**
* Boost the precision by appending decimal zeros to the value. This returns a value which appears to have
* a higher precision than the input.
*
* @param x The input value
* @param mc The requirement on the minimum precision on return.
* @return The same value as the input but with increased (pseudo) precision.
*/
static public BigDecimal scalePrec(final BigDecimal x, final MathContext mc) {
final int diffPr = mc.getPrecision() - x.precision();
if (diffPr > 0) {
return scalePrec(x, diffPr);
} else {
return x;
}
} /* BigDecimalMath.scalePrec */
/**
* Convert an absolute error to a precision.
*
* @param x The value of the variable
* @param xerr The absolute error in the variable
* @return The number of valid digits in x.
* The value is rounded down, and on the pessimistic side for that reason.
*/
static public int err2prec(BigDecimal x, BigDecimal xerr) {
return err2prec(xerr.divide(x, MathContext.DECIMAL64).doubleValue());
}
/**
* Convert an absolute error to a precision.
*
* @param x The value of the variable
* The value returned depends only on the absolute value, not on the sign.
* @param xerr The absolute error in the variable
* The value returned depends only on the absolute value, not on the sign.
* @return The number of valid digits in x.
* Derived from the representation x+- xerr, as if the error was represented
* 38
* in a "half width" (half of the error bar) form.
* The value is rounded down, and on the pessimistic side for that reason.
*/
static public int err2prec(double x, double xerr) {
/* Example: an error of xerr=+-0.5 at x=100 represents 100+-0.5 with
* a precision = 3 (digits).
*/
return 1 + (int) (Math.log10(Math.abs(0.5 * x / xerr)));
}
/**
* Convert a relative error to a precision.
*
* @param xerr The relative error in the variable.
* The value returned depends only on the absolute value, not on the sign.
* @return The number of valid digits in x.
* The value is rounded down, and on the pessimistic side for that reason.
*/
static public int err2prec(double xerr) {
/* Example: an error of xerr=+-0.5 a precision of 1 (digit), an error of
* +-0.05 a precision of 2 (digits)
*/
return 1 + (int) (Math.log10(Math.abs(0.5 / xerr)));
}
/**
* Convert a precision (relative error) to an absolute error.
* The is the inverse functionality of err2prec().
*
* @param x The value of the variable
* The value returned depends only on the absolute value, not on the sign.
* @param prec The number of valid digits of the variable.
* @return the absolute error in x.
* Derived from the an accuracy of one half of the ulp.
*/
static public double prec2err(final double x, final int prec) {
return 5. * Math.abs(x) * Math.pow(10., -prec);
}
} /* BigDecimalMath */
