MultiplyNum | optimized numerical multiplication |
CachedConstant | precompute multiple-precision constants |
NewtonNum | low-level optimized Newton's iterations |
SumTaylorNum | optimized numerical evaluation of Taylor series |
IntPowerNum | optimized computation of integer powers |
BinSplitNum | computations of series by the binary splitting method |
BinSplitData | computations of series by the binary splitting method |
BinSplitFinal | computations of series by the binary splitting method |
MathSetExactBits | manipulate precision of floating-point numbers |
MathGetExactBits | manipulate precision of floating-point numbers |
InNumericMode | determine if currently in numeric mode |
NonN | calculate part in non-numeric mode |
IntLog | integer part of logarithm |
IntNthRoot | integer part of n-th root |
NthRoot | calculate/simplify nth root of an integer |
ContFracList | manipulate continued fractions |
ContFracEval | manipulate continued fractions |
GuessRational | find optimal rational approximations |
NearRational | find optimal rational approximations |
BracketRational | find optimal rational approximations |
TruncRadian | remainder modulo 2*Pi |
Builtin'Precision'Set | set the precision |
Builtin'Precision'Get | get the current precision |
MultiplyNum(x,y) MultiplyNum(x,y,z,...) MultiplyNum({x,y,z,...}) |
The function accepts any number of arguments (not less than two) or a list of numbers. The result is always a floating-point number (even if InNumericMode() returns False).
CachedConstant(cache, Cname, Cfunc) |
Cname -- atom, name of the constant
Cfunc -- expression that evaluates the constant
The call to CachedConstant defines a new function named Cname() that returns the value of the constant at given precision. If the precision is increased, the value will be recalculated as necessary, otherwise calling Cname() will take very little time.
The parameter Cfunc must be an expression that can be evaluated and returns the value of the desired constant at the current precision. (Most arbitrary-precision mathematical functions do this by default.)
The associative list cache contains elements of the form {Cname, prec, value}, as illustrated in the example. If this list does not exist, it will be created.
This mechanism is currently used by N() to precompute the values of Pi and gamma (and the golden ratio through GoldenRatio, and Catalan). The name of the cache for N() is CacheOfConstantsN. The code in the function N() assigns unevaluated calls to Internal'Pi() and Internal'gamma() to the atoms Pi and gamma and declares them to be lazy global variables through SetGlobalLazyVariable (with equivalent functions assigned to other constants that are added to the list of cached constants).
The result is that the constants will be recalculated only when they are used in the expression under N(). In other words, the code in N() does the equivalent of
SetGlobalLazyVariable(mypi,Hold(Internal'Pi())); SetGlobalLazyVariable(mygamma,Hold(Internal'gamma())); |
After this, evaluating an expression such as 1/2+gamma will call the function Internal'gamma() but not the function Internal'Pi().
In> CachedConstant( my'cache, Ln2, Internal'LnNum(2) ) Out> True; In> Internal'Ln2() Out> 0.6931471806; In> V(N(Internal'Ln2(),20)) CachedConstant: Info: constant Ln2 is being recalculated at precision 20 Out> 0.69314718055994530942; In> my'cache Out> {{"Ln2",20,0.69314718055994530942}}; |
NewtonNum(func, x0, prec0, order) NewtonNum(func, x0, prec0) NewtonNum(func, x0) |
x0 -- initial value (must be close enough to the root)
prec0 -- initial precision (at least 4, default 5)
order -- convergence order (typically 2 or 3, default 2)
NewtonNum will iterate the given function starting from the initial value, until the sequence converges within current precision. Initially, up to 5 iterations at the initial precision prec0 is performed (the low precision is set for speed). The initial value x0 must be close enough to the root so that the initial iterations converge. If the sequence does not produce even a single correct digit of the root after these initial iterations, an error message is printed. The default value of the initial precision is 5.
The order parameter should give the convergence order of the scheme. Normally, Newton iteration converges quadratically (so the default value is order=2) but some schemes converge faster and you can speed up this function by specifying the correct order. (Caution: if you give order=3 but the sequence is actually quadratic, the result will be silently incorrect. It is safe to use order=2.)
In> Builtin'Precision'Set(20) Out> True; In> NewtonNum({{x}, x+Sin(x)}, 3, 5, 3) Out> 3.14159265358979323846; |
SumTaylorNum(x, NthTerm, order) SumTaylorNum(x, NthTerm, TermFactor, order) SumTaylorNum(x, ZerothTerm, TermFactor, order) |
ZerothTerm -- value of the 0-th coefficient of the series
x -- number, value of the expansion variable
TermFactor -- a function specifying the ratio of n-th term to the previous one
order -- power of x in the last term
The coefficients a[k] of the Taylor series are given as functions of one integer variable (k). It is convenient to pass them to SumTaylorNum as closures. For example, if a function a(k) is defined, then
SumTaylorNum(x, {{k}, a(k)}, n) |
Often a simple relation between successive coefficients a[k-1], a[k] of the series is available; usually they are related by a rational factor. In this case, the second form of SumTaylorNum should be used because it will compute the series faster. The function TermFactor applied to an integer k>=1 must return the ratio a[k]/a[k-1]. (If possible, the function TermFactor should return a rational number and not a floating-point number.) The function NthTerm may also be given, but the current implementation only calls NthTerm(0) and obtains all other coefficients by using TermFactor. Instead of the function NthTerm, a number giving the 0-th term can be given.
The algorithm is described elsewhere in the documentation. The number of terms order+1 must be specified and a sufficiently high precision must be preset in advance to achieve the desired accuracy. (The function SumTaylorNum does not change the current precision.)
In> Builtin'Precision'Set(21) Out> True; In> SumTaylorNum(1, {{k},1/k!}, 21) Out> 2.718281828459045235351; In> SumTaylorNum(1, 1, {{k},1/k}, 21) Out> 2.71828182845904523535; In> SumTaylorNum(1, {{k},1/k!}, {{k},1/k}, 21) Out> 2.71828182845904523535; In> RoundTo(N(Ln(%)),20) Out> 1; |
IntPowerNum(x, n, mult, unity) |
n -- a non-negative integer (power to raise x to)
mult -- a function that performs one multiplication
unity -- value of the unity with respect to that multiplication
Mathematically, this function is a generalization of MathPower to rings other than that of real numbers.
In the current implementation, the unity argument is only used when the given power n is zero.
In> IntPowerNum(3, 3, MathMultiply,1) Out> 27; |
In> IntPowerNum(3+4*I, 3, *,1) Out> Complex(-117,44); In> IntPowerNum(HilbertMatrix(2), 4, *, Identity(2)) Out> {{289/144,29/27},{29/27,745/1296}}; |
In> IntPowerNum(3,100,{{x,y},Mod(x*y,7)},1) Out> 4; |
BinSplitNum(n1, n2, a, b, c, d) BinSplitData(n1,n2, a, b, c, d) BinSplitFinal({P,Q,B,T}) |
a, b, c, d -- functions of one argument, coefficients of the series
P, Q, B, T -- numbers, intermediate data as returned by BinSplitData
The last four arguments of BinSplitNum are functions of one argument that give the coefficients a(k), b(k), p(k), q(k). In most cases these will be short integers that are simple to determine. The binary splitting method will work also for non-integer coefficients, but the calculation will take much longer in that case.
Note: the binary splitting method outperforms the straightforward summation only if the multiplication of integers is faster than quadratic in the number of digits. See the algorithm documentation for more information.
The two other functions are low-level functions that allow a finer control over the calculation. The use of the low-level routines allows checkpointing or parallelization of a binary splitting calculation.
The binary splitting method recursively reduces the calculation of S(n[1],n[2]) to the same calculation for the two halves of the interval [n[1], n[2]]. The intermediate results of a binary splitting calculation are returned by BinSplitData and consist of four integers P, Q, B, T. These four integers are converted into the final answer S by the routine BinSplitFinal using the relation
In> Builtin'Precision'Set(21) Out> True; In> BinSplitNum(1,21, {{k},1}, {{k},1},{{k},1},{{k},k}) Out> 1.718281828459045235359; In> N(Exp(1)-1) Out> 1.71828182845904523536; In> BinSplitData(1,21, {{k},1}, {{k},1},{{k},1},{{k},k}) Out> {1,51090942171709440000,1, 87788637532500240022}; In> BinSplitFinal(%) Out> 1.718281828459045235359; |
MathGetExactBits(x) MathSetExactBits(x,bits) |
bits -- integer, number of bits
MathGetExactBits(x) returns an integer number n such that x represents a real number in the interval [ x*(1-2^(-n)), x*(1+2^(-n))] if x!=0 and in the interval [-2^(-n), 2^(-n)] if x=0. The integer n is always nonnegative unless x is zero (a "floating zero"). A floating zero can have a negative value of the number n of exact bits.
These functions are only meaningful for floating-point numbers. (All integers are always exact.) For integer x, the function MathGetExactBits returns the bit count of x and the function MathSetExactBits returns the unmodified integer x.
In> MathGetExactBits(1000.123) Out> 33; In> x:=MathSetExactBits(10., 20) Out> 10.; In> MathGetExactBits(x) Out> 20; |
In> x:=MathSetExactBits(0., -2) Out> 0.; In> x=0 Out> True; |
NonN(expr) InNumericMode() |
prec -- integer, precision to use
InNumericMode() would typically be used to define a transformation rule that defines how to get a numeric approximation of some expression. One could define a transformation rule
f(_x)_InNumericMode() <- [... some code to get a numeric approximation of f(x) ... ]; |
InNumericMode() usually returns False, so transformation rules that check for this predicate are usually left alone.
When in numeric mode, NonN can be called to switch back to non-numeric mode temporarily.
NonN is a macro. Its argument expr will only be evaluated after the numeric mode has been set appropriately.
In> InNumericMode() Out> False In> N(InNumericMode()) Out> True In> N(NonN(InNumericMode())) Out> False |
IntLog(n, base) |
This function can also be used to quickly count the digits in a given number.
In> IntLog(257^8, 2) Out> 64; |
Count the number of decimal digits:
In> IntLog(321^321, 10) Out> 804; |
IntNthRoot(x, n) |
This function is used to test numbers for prime powers.
In> IntNthRoot(65537^111, 37) Out> 281487861809153; |
NthRoot(m,n) |
n -- a positive integer greater than 1 ( n>1)
For large m and small n NthRoot may work quite slowly. Every result {f,r} for given m, n is saved in a lookup table, thus subsequent calls to NthRoot with the same values m, n will be executed quite fast.
In> NthRoot(12,2) Out> {2,3}; In> NthRoot(81,3) Out> {3,3}; In> NthRoot(3255552,2) Out> {144,157}; In> NthRoot(3255552,3) Out> {12,1884}; |
ContFracList(frac) ContFracList(frac, depth) ContFracEval(list) ContFracEval(list, rest) |
depth -- desired number of terms
list -- a list of coefficients
rest -- expression to put at the end of the continued fraction
The function ContFracEval converts a list of coefficients into a continued fraction expression. The optional parameter rest specifies the symbol to put at the end of the expansion. If it is not given, the result is the same as if rest=0.
In> A:=ContFracList(33/7 + 0.000001) Out> {4,1,2,1,1,20409,2,1,13,2,1,4,1,1,3,3,2}; In> ContFracEval(Take(A, 5)) Out> 33/7; In> ContFracEval(Take(A,3), remainder) Out> 1/(1/(remainder+2)+1)+4; |
GuessRational(x) GuessRational(x, digits) NearRational(x) NearRational(x, digits) BracketRational(x, eps) |
digits -- desired number of decimal digits (integer)
eps -- desired precision
Unlike the function Rationalize() which converts floating-point numbers to rationals without loss of precision, the functions GuessRational() and NearRational() are intended to find the best rational that is approximately equal to a given value.
The function GuessRational() is useful if you have obtained a floating-point representation of a rational number and you know approximately how many digits its exact representation should contain. This function takes an optional second parameter digits which limits the number of decimal digits in the denominator of the resulting rational number. If this parameter is not given, it defaults to half the current precision. This function truncates the continuous fraction expansion when it encounters an unusually large value (see example). This procedure does not always give the "correct" rational number; a rule of thumb is that the floating-point number should have at least as many digits as the combined number of digits in the numerator and the denominator of the correct rational number.
The function NearRational(x) is useful if one needs to approximate a given value, i.e. to find an "optimal" rational number that lies in a certain small interval around a certain value x. This function takes an optional second parameter digits which has slightly different meaning: it specifies the number of digits of precision of the approximation; in other words, the difference between x and the resulting rational number should be at most one digit of that precision. The parameter digits also defaults to half of the current precision.
The function BracketRational(x,eps) can be used to find approximations with a given relative precision from above and from below. This function returns a list of two rational numbers {r1,r2} such that r1<x<r2 and Abs(r2-r1)<Abs(x*eps). The argument x must be already evaluated to enough precision so that this approximation can be meaningfully found. If the approximation with the desired precision cannot be found, the function returns an empty list.
In> x:=N(956/1013) Out> 0.9437314906 In> Rationalize(x) Out> 4718657453/5000000000; In> V(GuessRational(x)) GuessRational: using 10 terms of the continued fraction Out> 956/1013; In> ContFracList(x) Out> {0,1,16,1,3,2,1,1,1,1,508848,3,1,2,1,2,2}; |
In> NearRational(x) Out> 218/231; |
In> NearRational(x, 10) Out> 956/1013; |
In> BracketRational(N(Ln(10)), 10^(-8)) Out> {12381/5377,41062/17833}; |
TruncRadian(r) |
The library uses the formula
In> 2*Internal'Pi() Out> 6.283185307; In> TruncRadian(6.28) Out> 6.28; In> TruncRadian(6.29) Out> 0.0068146929; |
Builtin'Precision'Set(n) |
This is not the number of digits after the decimal point. For example, 123.456 has 3 digits after the decimal point and 6 digits of mantissa. The number 123.456 is adequately computed by specifying Builtin'Precision'Set(6).
The call Builtin'Precision'Set(n) will not guarantee that all results are precise to n digits.
When the precision is changed, all variables containing previously calculated values remain unchanged. The Builtin'Precision'Set function only makes all further calculations proceed with a different precision.
Also, when typing floating-point numbers, the current value of Builtin'Precision'Set is used to implicitly determine the number of precise digits in the number.
In> Builtin'Precision'Set(10) Out> True; In> N(Sin(1)) Out> 0.8414709848; In> Builtin'Precision'Set(20) Out> True; In> x:=N(Sin(1)) Out> 0.84147098480789650665; |
The value x is not changed by a Builtin'Precision'Set() call:
In> [ Builtin'Precision'Set(10); x; ] Out> 0.84147098480789650665; |
The value x is rounded off to 10 digits after an arithmetic operation:
In> x+0. Out> 0.8414709848; |
In the above operation, 0. was interpreted as a number which is precise to 10 digits (the user does not need to type 0.0000000000 for this to happen). So the result of x+0. is precise only to 10 digits.
Builtin'Precision'Get() |
In> Builtin'Precision'Get(); Out> 10; In> Builtin'Precision'Set(20); Out> True; In> Builtin'Precision'Get(); Out> 20; |