See: Description
Interface | Description |
---|---|
DoubleConstants |
Constants needed for various algorithms for the
double type. |
DoubleModConstants |
Constants needed for various modular arithmetic operations for the
double type. |
DoubleRadixConstants |
Constants related to different radixes for the
double data type. |
FloatConstants |
Constants needed for various algorithms for the
float type. |
FloatModConstants |
Constants needed for various modular arithmetic operations for the
float type. |
FloatRadixConstants |
Constants related to different radixes for the
float data type. |
IntConstants |
Constants needed for various algorithms for the
int type. |
IntModConstants |
Constants needed for various modular arithmetic operations for the
int type. |
IntRadixConstants |
Constants related to different radixes for the
int data type. |
LongConstants |
Constants needed for various algorithms for the
long type. |
LongModConstants |
Constants needed for various modular arithmetic operations for the
long type. |
LongRadixConstants |
Constants related to different radixes for the
long data type. |
Parallelizable |
Any task that can use a
ParallelRunner to execute operations in parallel. |
Class | Description |
---|---|
AbstractConvolutionBuilder |
Abstract base class for creating convolutions of suitable type for the specified length.
|
AbstractDataStorageBuilder |
Abstract base class for a data storage creation strategy.
|
AbstractNTTBuilder |
Abstract base class for creating Number Theoretic Transforms suitable for the
specified length, based on available memory configured in the
ApfloatContext . |
AbstractStepFNTStrategy |
Abstract superclass for step-based FNT strategies.
|
DiskDataStorage |
Abstract base class for disk-based data storage, containing the common
functionality independent of the element type.
|
DoubleAdditionBuilder |
Creates additions for the specified radix and the
double element type. |
DoubleAdditionStrategy |
Basic addition strategy for the
double element type. |
DoubleApfloatBuilder |
Builder class for building
ApfloatImpl implementations with the
double data element type. |
DoubleApfloatImpl |
Immutable apfloat implementation class for the
double data element type. |
DoubleBaseMath |
Mathematical operations on numbers in a base.
|
DoubleBuilderFactory |
Factory class for getting instances of the various builder classes needed
to build an
ApfloatImpl with the double data element type. |
DoubleCarryCRTBuilder |
Creates carry-CRT related objects, for the
double type. |
DoubleCarryCRTStepStrategy |
Class for performing the final steps of a three-modulus
Number Theoretic Transform based convolution.
|
DoubleConvolutionBuilder |
Creates convolutions of suitable type for the
double type. |
DoubleCRTMath |
Basic arithmetic for calculating the Chinese Remainder
Theorem.
|
DoubleDataStorageBuilder |
Default data storage creation strategy for the
double data type. |
DoubleDiskDataStorage |
Disk-based data storage for the
double element type. |
DoubleElementaryModMath |
Elementary modulo arithmetic functions for
double data. |
DoubleFactor3NTTStepStrategy |
Steps for the factor-3 NTT.
|
DoubleKaratsubaConvolutionStrategy |
Convolution strategy using the Karatsuba algorithm.
|
DoubleMatrixBuilder |
Creates matrix operations objects, for the
double type. |
DoubleMatrixStrategy |
Optimized matrix transposition methods for the
double type. |
DoubleMediumConvolutionStrategy |
Medium-length convolution strategy.
|
DoubleMemoryArrayAccess |
Array access class based on a
double[] . |
DoubleMemoryDataStorage |
Memory based data storage implementation for the
double
element type. |
DoubleModMath |
Modulo arithmetic functions for
double data. |
DoubleNTTBuilder |
Creates Number Theoretic Transforms for the
double type. |
DoubleNTTConvolutionStepStrategy |
Steps of a three-NTT convolution for the
double type. |
DoubleNTTStepStrategy |
Common methods to calculate Fast Number Theoretic Transforms
in parallel using multiple threads.
|
DoubleScramble |
Functions to perform bit-reverse ordering of
double data. |
DoubleShortConvolutionStrategy |
Short convolution strategy.
|
DoubleTableFNT |
Fast Number Theoretic Transform that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
DoubleTableFNTStrategy |
Fast Number Theoretic Transform strategy that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
DoubleWTables |
Helper class for generating and caching tables of powers of the n:th root of unity.
|
Factor3NTTStrategy |
A transform that implements a 3-point transform on
top of another Number Theoretic Transform that does
transforms of length 2n.
|
FloatAdditionBuilder |
Creates additions for the specified radix and the
float element type. |
FloatAdditionStrategy |
Basic addition strategy for the
float element type. |
FloatApfloatBuilder |
Builder class for building
ApfloatImpl implementations with the
float data element type. |
FloatApfloatImpl |
Immutable apfloat implementation class for the
float data element type. |
FloatBaseMath |
Mathematical operations on numbers in a base.
|
FloatBuilderFactory |
Factory class for getting instances of the various builder classes needed
to build an
ApfloatImpl with the float data element type. |
FloatCarryCRTBuilder |
Creates carry-CRT related objects, for the
float type. |
FloatCarryCRTStepStrategy |
Class for performing the final steps of a three-modulus
Number Theoretic Transform based convolution.
|
FloatConvolutionBuilder |
Creates convolutions of suitable type for the
float type. |
FloatCRTMath |
Basic arithmetic for calculating the Chinese Remainder
Theorem.
|
FloatDataStorageBuilder |
Default data storage creation strategy for the
float data type. |
FloatDiskDataStorage |
Disk-based data storage for the
float element type. |
FloatElementaryModMath |
Elementary modulo arithmetic functions for
float data. |
FloatFactor3NTTStepStrategy |
Steps for the factor-3 NTT.
|
FloatKaratsubaConvolutionStrategy |
Convolution strategy using the Karatsuba algorithm.
|
FloatMatrixBuilder |
Creates matrix operations objects, for the
float type. |
FloatMatrixStrategy |
Optimized matrix transposition methods for the
float type. |
FloatMediumConvolutionStrategy |
Medium-length convolution strategy.
|
FloatMemoryArrayAccess |
Array access class based on a
float[] . |
FloatMemoryDataStorage |
Memory based data storage implementation for the
float
element type. |
FloatModMath |
Modulo arithmetic functions for
float data. |
FloatNTTBuilder |
Creates Number Theoretic Transforms for the
float type. |
FloatNTTConvolutionStepStrategy |
Steps of a three-NTT convolution for the
float type. |
FloatNTTStepStrategy |
Common methods to calculate Fast Number Theoretic Transforms
in parallel using multiple threads.
|
FloatScramble |
Functions to perform bit-reverse ordering of
float data. |
FloatShortConvolutionStrategy |
Short convolution strategy.
|
FloatTableFNT |
Fast Number Theoretic Transform that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
FloatTableFNTStrategy |
Fast Number Theoretic Transform strategy that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
FloatWTables |
Helper class for generating and caching tables of powers of the n:th root of unity.
|
IntAdditionBuilder |
Creates additions for the specified radix and the
int element type. |
IntAdditionStrategy |
Basic addition strategy for the
int element type. |
IntApfloatBuilder |
Builder class for building
ApfloatImpl implementations with the
int data element type. |
IntApfloatImpl |
Immutable apfloat implementation class for the
int data element type. |
IntBaseMath |
Mathematical operations on numbers in a base.
|
IntBuilderFactory |
Factory class for getting instances of the various builder classes needed
to build an
ApfloatImpl with the int data element type. |
IntCarryCRTBuilder |
Creates carry-CRT related objects, for the
int type. |
IntCarryCRTStepStrategy |
Class for performing the final steps of a three-modulus
Number Theoretic Transform based convolution.
|
IntConvolutionBuilder |
Creates convolutions of suitable type for the
int type. |
IntCRTMath |
Basic arithmetic for calculating the Chinese Remainder
Theorem.
|
IntDataStorageBuilder |
Default data storage creation strategy for the
int data type. |
IntDiskDataStorage |
Disk-based data storage for the
int element type. |
IntElementaryModMath |
Elementary modulo arithmetic functions for
int data. |
IntFactor3NTTStepStrategy |
Steps for the factor-3 NTT.
|
IntKaratsubaConvolutionStrategy |
Convolution strategy using the Karatsuba algorithm.
|
IntMatrixBuilder |
Creates matrix operations objects, for the
int type. |
IntMatrixStrategy |
Optimized matrix transposition methods for the
int type. |
IntMediumConvolutionStrategy |
Medium-length convolution strategy.
|
IntMemoryArrayAccess |
Array access class based on a
int[] . |
IntMemoryDataStorage |
Memory based data storage implementation for the
int
element type. |
IntModMath |
Modulo arithmetic functions for
int data. |
IntNTTBuilder |
Creates Number Theoretic Transforms for the
int type. |
IntNTTConvolutionStepStrategy |
Steps of a three-NTT convolution for the
int type. |
IntNTTStepStrategy |
Common methods to calculate Fast Number Theoretic Transforms
in parallel using multiple threads.
|
IntScramble |
Functions to perform bit-reverse ordering of
int data. |
IntShortConvolutionStrategy |
Short convolution strategy.
|
IntTableFNT |
Fast Number Theoretic Transform that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
IntTableFNTStrategy |
Fast Number Theoretic Transform strategy that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
IntWTables |
Helper class for generating and caching tables of powers of the n:th root of unity.
|
LongAdditionBuilder |
Creates additions for the specified radix and the
long element type. |
LongAdditionStrategy |
Basic addition strategy for the
long element type. |
LongApfloatBuilder |
Builder class for building
ApfloatImpl implementations with the
long data element type. |
LongApfloatImpl |
Immutable apfloat implementation class for the
long data element type. |
LongBaseMath |
Mathematical operations on numbers in a base.
|
LongBuilderFactory |
Factory class for getting instances of the various builder classes needed
to build an
ApfloatImpl with the long data element type. |
LongCarryCRTBuilder |
Creates carry-CRT related objects, for the
long type. |
LongCarryCRTStepStrategy |
Class for performing the final steps of a three-modulus
Number Theoretic Transform based convolution.
|
LongConvolutionBuilder |
Creates convolutions of suitable type for the
long type. |
LongCRTMath |
Basic arithmetic for calculating the Chinese Remainder
Theorem.
|
LongDataStorageBuilder |
Default data storage creation strategy for the
long data type. |
LongDiskDataStorage |
Disk-based data storage for the
long element type. |
LongElementaryModMath |
Elementary modulo arithmetic functions for
long data. |
LongFactor3NTTStepStrategy |
Steps for the factor-3 NTT.
|
LongKaratsubaConvolutionStrategy |
Convolution strategy using the Karatsuba algorithm.
|
LongMatrixBuilder |
Creates matrix operations objects, for the
long type. |
LongMatrixStrategy |
Optimized matrix transposition methods for the
long type. |
LongMediumConvolutionStrategy |
Medium-length convolution strategy.
|
LongMemoryArrayAccess |
Array access class based on a
long[] . |
LongMemoryDataStorage |
Memory based data storage implementation for the
long
element type. |
LongModMath |
Modulo arithmetic functions for
long data. |
LongNTTBuilder |
Creates Number Theoretic Transforms for the
long type. |
LongNTTConvolutionStepStrategy |
Steps of a three-NTT convolution for the
long type. |
LongNTTStepStrategy |
Common methods to calculate Fast Number Theoretic Transforms
in parallel using multiple threads.
|
LongScramble |
Functions to perform bit-reverse ordering of
long data. |
LongShortConvolutionStrategy |
Short convolution strategy.
|
LongTableFNT |
Fast Number Theoretic Transform that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
LongTableFNTStrategy |
Fast Number Theoretic Transform strategy that uses lookup tables
for powers of n:th root of unity and permutation indexes.
|
LongWTables |
Helper class for generating and caching tables of powers of the n:th root of unity.
|
MessagePasser<K,V> |
Message passing helper class for parallel codes.
|
ParallelRunnable |
Abstract class for a
Runnable that can be run in parallel by
multiple threads. |
ParallelRunner |
Class for running
ParallelRunnable objects in parallel using
multiple threads. |
ParallelThreeNTTConvolutionStrategy |
Convolution using three Number Theoretic Transforms
and the CRT to get the final result, using multiple threads in parallel.
|
Scramble |
Functions to perform bit-reverse ordering of data.
|
SixStepFNTStrategy |
Fast Number Theoretic Transform that uses a "six-step"
algorithm to calculate a long transform more efficiently on
cache-based memory architectures.
|
StepCarryCRTStrategy |
Class for performing the final step of a three-modulus
Number Theoretic Transform based convolution.
|
ThreeNTTConvolutionStrategy |
Convolution using three Number Theoretic Transforms
and the Chinese Remainder Theorem to get the final result.
|
TwoPassFNTStrategy |
Fast Number Theoretic Transform that uses a "two-pass"
algorithm to calculate a very long transform on data that
resides on a mass storage device.
|
Exception | Description |
---|---|
ApfloatInternalException |
Exception indicating some unexpected apfloat
implementation specific error situation.
|
BackingStorageException |
Exception indicating a backing storage failure.
|
ImplementationMismatchException |
Exception indicating a different implementation of the apfloat SPI
being used in two operands of a calculation.
|
RadixMismatchException |
Exception indicating a different radix being used in two operands
of a calculation.
|
TransformLengthExceededException |
Exception indicating that the "size" of the numbers used in a
multiplication is too large.
|
The org.apfloat.internal
package contains four different
implementations of the apfloat SPI, each based on a different primitive
element type:
IntBuilderFactory
, based on element type
int
: This is the default implementation used by apfloat.
It works well for 32-bit platforms that perform integer operations fast
(including integer multiplication), and can multiply double
s
and convert between double
and int
with adequate
performance. This applies to most workstations today (Intel x86 processors
and compatibles, in particular processors with SSE2 support, and most RISC
architectures). You can do calculations up to roughly 226 million digits
(in radix 10) with this implementation, which should be enough for most
purposes.LongBuilderFactory
, based on element type
long
: This implementation uses the 64-bit long
integer as the elementary type for all data storage and manipulation. It
usually is faster than the int
version on 64-bit architectures
if you have a JVM that actually uses the 64-bit features of the processor.
In some places it uses also double
arithmetic, so the processor
should be able to perform double-precision floating point operations as well
as convert between double
and long
, for decent
performance. For example, on x86-64 and SPARC the 64-bit long
version is faster than the 32-bit int
version. You can use the
long
implementation on 32-bit platforms too, however the
performance per element is less than half of the int
version,
even if roughly twice as much data is processed per element. The upside
is that this implementation can do much bigger calculations: up to about
3.5 * 1015 digits in radix 10.DoubleBuilderFactory
, based on element type
double
: This implementation exists generally only as a
curiosity. It will typically perform worse than the long
version, and it's only able to do calculations with about 1/20 of its
maximum digit length. The only situation where using the double
version might make sense is on a platform that performs floating-point
arithmetic well, but performs integer arithmetic extremely badly. Finding
such a platform today might be difficult, so generally it's advisable to
use the long
version instead, if you have a 64-bit platform
or need the most extreme precision.FloatBuilderFactory
, based on element type
float
: This version is also only a curiosity. The main
downside is that it can only perform calculations up to about 1.3
million radix-10 digits. The per-digit performance is also typically
less than that of the int
version. Unless you have a
computer that performs floating-point arithmetic extraordinarily well
compared to integer arithmetic, it's always advisable to use the
int
version instead.Type | Pentium 4 | Athlon XP | Athlon 64 (32-bit) | Athlon 64 (64-bit) | UltraSPARC II |
---|---|---|---|---|---|
Int | 100% | 100% | 100% | 100% | 100% |
Long | 40% | 76% | 59% | 95% | 132% |
Double | 45% | 63% | 59% | 94% | 120% |
Float | 40% | 43% | 46% | 42% | 82% |
Compared to the java.math.BigInteger
class with different digit
sizes, the apfloat relative performance with the same CPUs is as follows:
(Test was done with apfloat 1.1 using Sun's Java 5.0 server VM calculating 3n and converting the result to decimal.)
This benchmark suggests that for small numbers less than roughly 200 decimal
digits in size the BigInteger
/ BigDecimal
classes
are probably faster, even by an order of magnitude. Using apfloats is only beneficial
for numbers that have at least a couple hundred digits, or of course if some
mathematical functions are needed that are not available for BigInteger
s
or BigDecimal
s. The results can be easily explained by the smaller overhead
that BigInteger
s have due to their simpler implementation. When the size
of the mantissa grows, the O(n log n) complexity of apfloat's FFT-based multiplication
makes apfloat considerably faster than the steady O(n2) implementation
of the BigInteger
class. For numbers with millions of digits,
multiplication using BigInteger
s would be simply unfeasible, whereas for
apfloat it would not be a problem at all.
All of the above apfloat implementations have the following features (some of the links
point to the int
version, but all four versions have similar classes):
IntMemoryDataStorage
) or on disk
(IntDiskDataStorage
).IntShortConvolutionStrategy
),
using a simple O(n2) long multiplication algorithm for small numbers,
with low overhead (IntMediumConvolutionStrategy
),
using the Karatsuba multiplication algorithm for slightly larger numbers,
with some more overhead (IntKaratsubaConvolutionStrategy
),
or using a Number Theoretic Transform (NTT) done using three different moduli,
and the final result calculated using the Chinese Remainder Theorem
(ThreeNTTConvolutionStrategy
), for big numbers.IntTableFNTStrategy
) when the entire transform
fits in the processor cache, "six-step" NTT when the transform fits in the
main memory (SixStepFNTStrategy
),
and a disk-based "two-pass" NTT strategy when the whole transform doesn't
fit in the available memory (TwoPassFNTStrategy
).ApfloatInternalException
.
This exception, or various subclasses can be thrown in different situations, for
example:
IOException
can be thrown in any of the disk operations,
if e.g. a file can't be created, or written to if the disk is full.*.ap
and they are by default created in the current working directory. When the objects
are garbage collected, the temporary files are deleted. However, garbage collection
may not work perfectly at all times, and in general there are no guarantees that
it will happen at all. So, depending on the program being executed, it may be
beneficial to explicitly call System.gc()
at some point to ensure
that unused temporary files are deleted. However, VM vendors generally warn
against doing this too often, since it may seriously degrade performance. So,
figuring out how to optimally call it may be difficult. If the file deletion fails
for some reason, some temporary files may be left on disk after the program
exits. These files can be safely removed after the program has terminated.Many parts of the program are parallelized i.e. are processed with multiple threads in parallel. Parallelization is done where it has been easy to implement and where it is efficient. E.g. the "six-step" NTT is parallelized, because the data is in matrix form in memory and it's easy and highly efficient to process the rows of the matrix in parallel. Other places where parallelization is implemented are the in-place multiplication of transform results and the carry-CRT operation. However in both of these algorithms the process is parallelized only if the data is in memory - if the data was stored on disk then the irregular disk seeking could make the parallel algorithm highly inefficient.
Many sections of the code are not parallelized, where it's obvious that parallelization would not bring any benefits. Examples of such cases are addition, subtraction and matrix transposition. While parallel algorithms for these operations could certainly be implemented, they would not bring any performance improvement. The bottleneck in these operations is memory or I/O bandwidth and not CPU processing time. The CPU processing in addition and subtraction is highly trivial; in matrix transposition it's outright nonexistent - the algorithm only moves data from one place to another. Even if all the data was stored in memory, the memory bandwidth would be the bottleneck. E.g. in addition, the algorithm only needs a few CPU cycles per element to be processed. However moving the data from main memory to CPU registers and back to main memory needs likely significantly more CPU cycles than the addition operation itself. Parallelization would therefore not improve efficiency at all - the total CPU load might appear to increase but when measured in wall-clock time the execution would not be any faster.
Since the core functionality of the apfloat implementation is based on the
original C++ version of apfloat, no significant new algorithms have been
added (although the architecture has been otherwise greatly beautified e.g. by
separating the different implementations behind a SPI, and applying all kinds
of patterns everywhere). Thus, there are no different implementations for e.g.
using a floating-point FFT instead of a NTT, as the SPI (org.apfloat.spi
)
might suggest. However the default implementation does implement all the
patterns suggested by the SPI in fact the SPI was designed for the
default implementation.
The class diagram for an example apfloat that is stored on disk is shown below.
Note that all the aggregate classes can be shared by multiple objects that point
to the same instance. For example, multiple Apfloats can point to the same
ApfloatImpl, multiple ApfloatImpls can point to the same DataStorage etc. This
sharing happens in various situations, e.g. by calling floor()
,
multiplying by one etc:
The sequence diagram for creating a new apfloat that is stored on disk is as follows. Note that the FileStorage class is a private inner class of the DiskDataStorage class:
The sequence diagram for multiplying two apfloats is as follows. In this case a NTT based convolution is used, and the resulting apfloat is stored in memory:
Most of the files in the apfloat implementations are generated from templates
where a template tag is replaced by int/long/float/double
or
Int/Long/Float/Double
. Also the byte size of the element type is
templatized and replaced by 4/8/4/8. The only files that are individually
implemented for each element type are:
BaseMath.java CRTMath.java ElementaryModMath.java ModConstants.java
org.apfloat.spi
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