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, BinSplitData, BinSplitFinal | computations of series by the binary splitting method |
MathSetExactBits, MathGetExactBits | manipulate precision of floating-point numbers |
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 Numeric is not set).
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. The name of the cache for N() is CacheOfConstantsN. The code in the function N() assigns unevaluated calls to Pi() and gamma() to the atoms Pi and gamma and declares them LazyGlobal. 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
Pi := Hold(Pi()); gamma := Hold(gamma()); LazyGlobal(Pi); LazyGlobal(gamma); |
In> CachedConstant( my'cache, Ln2, LnNum(2) ) Out> 0.6931471806; In> Ln2 Out> Ln2; In> Ln2() Out> 0.6931471806; In> [ Precision(20); V( Ln2() ); ] 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> Precision(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> Precision(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> Precision(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; |