??????? ?? ???????????? ??????????? ?? Java

1
???13
??????? ?? ???????????? ??????????? ?? Java
?. ?. ??????? ??? ????? ??????????? ???
?????-?????????, ?????? dmitry.nadezhin_at_oracle.co
m
?. ?. ????? ????????? ???????????????
??????????? ???????, ?????? sergei_at_asu.ru
2
????

• ???????? ? ???????????? ??????????
• ?????? ????????? P1788 ???????? ????
• ????????? ? ????????? ??
• ?????????? ????????? (???????? ???????)
• ????????? ????, ?????????, ????????
• ??????? ? ???????
• ?????
• ??????

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3
?????? ????????? P1788

• ?????? ???????????? 1, 2, 3, 4
• ????? ?????????-?????????????, ???????
• ???????????? ????????? ????? ??? ???? ??????
• ??????
• ???????????? ??? (??????? 2)

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4
?????? ????????????

• 1 ??????? ?????????????? ??????
• 2 ??????? ???????????? ????? ??????
• 3 ??????? ?????????????
• 4 ??????? ??????? ?????

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5
???????????? ?????????

• a,b, a lt b

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?????????-????????????? ?????

• a,b, a lt b
• Ø
• -8,b
• a,8
• -8,8

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7
????? ???????

• a,b, a lt b
• a,b, a gt b

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??????

• f(x) Dom(f)
• Com x - ???????, x ? Dom(f) , f(x) ?????????? ?
  ?????? ????? ????????? x ? x, ?????????
  ???????????? ????????
• Dac x - ???????, x ? Dom(f) , ??????????? f(x)
  ?? x ??????????
• Def x - ???????, x ? Dom(f)
• Trv ?????? true
• Ill ?? ???????? NaI, ????????? Dom(f) ????

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??????

• f(x) 1/x Dom(f)-8,0) ? (0,8
• f(1,2_com) ½,1_com
• f(1,8_com) 0,1_dac
• sign(0,½_com)0,1_def
• f(0,1_com) 1,8_trv
• f(0,0_com) empty_trv
• nums2interval(2,1) empty_ill

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???????????? ??? (??????? 2)

• ???????? ??????????? ?????????? ??????????????
  ??????????
• ??????? hull ???????? ?? ???? ??????????????
  ?????????? ? ????????? ????????????

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11
????????? ? ????????? ??

• Java 7.0u25 ? NetBeans 7.3
• Ant ??? Maven
• ??????? Maven - ??????????? /.m2
• ????? ????????? ???????? ????? ?????????
  ltofflinegtfalselt/offlinegt
• ? ????? /.m2/settings.xml
• ?? ??????????? ???????? ?????? Jinterval
• /NetBeansProjects/net.java.jintervalsvn

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???????? ???????

• FileNew Project...
• Choose ProjectMavenJava ApplicationNextgt
• Project name tutor-rational
• GroupId ru.asu
• Package ru.asu.tutor.rational
• Enter
• Open ru.asu.tutor.rational.App

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NetBeans ?? ????? ??? Rational
package ru.asu.tutor.rational / Hello
world! / public class App public
static void main( String args )
Rational rats Rational.zero()
System.out.println( "Hello
World!" )
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?????????? ????????????

• ???????? ? Projects ????
• tutor-rationalDependencies
• ?????? ?????? ??????? ????
• Add dependency
• Query rational
• Unfold net.java.jintervaljinterval-rational-java
  
• Choose 0.1-SNAPSHOT jar local
• ? ???? ?????????????? ?????? ?????? ????
  FixImport

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?????? ????????
package ru.asu.tutor.rational import
net.java.jinterval.rational.Rational /
Hello world! / public class App
public static void main( String args )
Rational rats
Rational.zero()
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??????? ???????? Rational
package ru.asu.tutor.rational import
net.java.jinterval.rational.Rational /
Hello world! / public class App
public static void main( String args )
Rational rats
Rational.zero(), Rational.valueOf(2),
Rational.valueOf(0.1),
Rational.valueOf(1, 10) for
(Rational r rats)
System.out.println(r)
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??????? ?????? Rational
MathContext mc20f new MathContext(20,
RoundingMode.FLOOR) MathContext mc20c
new MathContext(20, RoundingMode.CEILING)
for (Rational r rats)
System.out.println(r " "
r.doubleValue() " "
r.bigDecimalValue(mc20f)
"," r.bigDecimalValue(mc20c)
"") 0.0 0.0 0,0 0x1p1 2.0
2,2 0xccccccccccccdp-55 0.1 0.1000000000000000
0555,0.10000000000000000556 0x1/0x52-1 0.1
0.1,0.1
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?????? ???????? ??? Rational
ExtendedRational diff
ExtendedRationalOps.subtract(rats2, rats3)
System.out.println("diff " diff
" " diff.doubleValue()
" " diff.bigDecimalValue(mc20f)
"," diff.bigDecimalValue(mc20c)
"") diff 0x1/0x52-55
5.551115123125783E-18 5.5511151231257827021E-18,5
.5511151231257827022E-18
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Javadoc ???, ???????? ?? ?????????

• FileOpen Project...
• /NetBeansProject/net.java.jintervalsvn/trunk/jin
  terval
• ? Projects ???????? jintervalModules
• ???????? ?????? ??????????? jinterval
• ??????? ?????? ?? jinterval-rational-java
• ???????? jinterval-rational-javaSource packages
• ???????? ?????? ???????
• ???????? net.java.jinterval.rational
• ??????? ExtendedRationalContext

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Rump Example in Rationals

• FileOpen Project...
• /NetBeansProjects/RumpExample

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JInterval Library Principles, Development, and
Perspectives
WHY INTERVAL COMPUTATIONS IN JVM?
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Java Is Popular

• TIOBE Programming Community Index for September
  2012
• Calculated by counting hits of the most popular
  search engines

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Java Is Popular

• RedMonks language ranking for September 2012

Popularity Rank on StackOverflow.com (by of
tags)
Popularity Rank on Github.com (by of projects)
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Java Is Popular

•  Bookscan's reports on the top 3,000 titles sold

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Gap between Interval and Applied Software

• Java is an attractive and widely adopted
  technology for applied software development
• Cross-platform portability of applications
• General purpose object-oriented language
• Almost any language can generate Java bytecodes
• Advanced tools for distributed systems
  development
• Huge amount of applied libraries
• Interval analysis and interval computations have
  proved to be useful in numerous real-world
  applications
• Interval software in Java is of fragmentary
  character
• Creation of systematic full-featured high-level
  interval library for Java brings interval tools
  closer to developers of applied software

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Interval Computations in Java

• Is Java suitable for scientific computing?

• Pro
• Portability of Java Virtual Machine (JVM)
• Safe memory management(no memory leaks and
  pointer errors)
• Network-aware environment
• Parallel and distributed computing (threads,
  RMI)
• Strict model of security
• Standard API for GUI, graphics, DBMS,
• Widely adopted
• Embedded systems, browsers,
• Development, teaching,

• Con
• Low performance
• Virtual machine
• Interpretation is slow
• Overhead cost of safe memory management
• Language restrictions
• No primitive structure type
• No operator overloading
• No traditional multidimesional arrays
• No full compliance with IEEE 754
• Relatively small number of scientific libraries
  on Java
• Scientific computing traditions Fortran, ?/?

Kahan W., Darcy J.D. How Javas Floating-Point
Hurts Everyone Everywhere//ACM 1998 Workshop on
Java for HighPerformance Network Computing,
Stanford University, March 1998,
http//www.cs.berkeley.edu/wkahan/JAVAhurt.pdf
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Interval Java Libraries

• IA_math, 1997
• Classic IA, classic interval elementary functions
• Timothy J. Hickey,
• Brandeis University, Boston, USA
• interval.sourceforge.net/interval/
• Java-XSC, 1999
• Classic IA, rectangular complex IA, classic
  interval elementary functions, classic and
  complex interval vectors and matrices
• Benjamin R.C. Bedregal, Jose E.M. Dutra
• Universidade Federal do Rio Grande do Norte,
  Natal, Brazil
• www.dimap.ufrn.br/java-xsc/jxsc2007.html

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JInterval Library Principles, Development, and
Perspectives
JINTERVAL EVOLUTION
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Stages of JInterval Evolution

• Sep 2008 JInterval is started as undergraduate
  student project Childhood at Altai State
  University (Barnaul, Russia) http//code.google.c
  om/p/javaintervalmathasu/
• Aug 2009 Dmitry Nadezhin (Sun Labs, Zelenograd,
  Russia) Boyhood joins the project http//kenai
  .com/projects/jinterval
• Jan 2012 Developing reference implementation and
  Youth test suite for P1788 becomes Priority 1
  http//java.net/projects/jinterval
  or http//jinterval.java.net

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JInterval (Boyhood) A Priori Requirements

• The library
• Must be clear and easy to use
• Should provide flexibility in the choice of
  interval algebra for computations
• Should provide flexibility to extend its
  functionality
• Should provide flexibility in choosing precision
  of interval boundaries and associated rounding
  policies
• Must be portable
• Should provide high performance
• Must be open source

Priority
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JInterval (Boyhood) Architecture

• Fast branch
• Interval bounds double, nearest rounding
• IA set-based, Kaucher, complex rectangular,
  complex circular, complex ring, complex polar
• Interval elementary functions, vectors, matrices
• ILS Gauss, Gauss-Seidel, subdifferential Newton,
  NonNeg, Shaidurov
• Rational bounds branch
• Interval bounds smart rational/double, arbitrary
  precision, rounding policies, contexts
• IA set-based
• Interval elementary functions, vectors, matrices
• Generic interfaces on top of branches

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JInterval (Boyhood) Type Hierarchy
Interval
ClassicRealInterval
ComplexInterval
RealInterval
DoubleInterval
ComplexIntervalCircle
RationalInterval
ComplexIntervalRectangle
ComplexIntervalPolar
ComplexIntervalRing
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JInterval (Boyhood) Lessons Learned
Problem Possible Solution
Java syntax is not expressive enough for calculations Developing Scala API for a new JInterval implementation
JInterval is not compliant with the project of interval standard IEEE P1788 Redesign the library according to IEEE P1788
Low performance Using optional plugins for native code of high precision arithmetic and interval linear algebra algorithms through JNA
Java r x.add(y.multiply(z))
Scala r x yz
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JInterval Library Principles, Development, and
Perspectives
ARCHITECTURE
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Class Diagram (package net.java.jinterval.interval
)
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Key-role Interfaces

• Types graph follows the flavor structure of P1788
• Java interfaces
• Interval
• Common methods for all flavors
• SetInterval
• Extends Interval with methods for flavor Set
  Interval
• KaucherInterval
• Extends Interval with methods for flavor Kaucher
  interval
• ClassicInterval
• Extends all flavors, because can be mapped to
  related flavor intervals

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Interface Interval common methods of all flavors

• Interval defines common methods of all interval
  flavors
• Defines numerical and boolean operations only

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IntervalContext interval operations interface

• Generic interface IntervalContext defines
  signature for interval-valued methods

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KaucherIntervalContext interval flavor interface

• KaucherIntervalContext extends
  IntervalContextand binds type variable I to
  Kaucher interval flavor

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Implementation of interval contexts

• There may be several implementations for flavor
  contexts
• SetIntervalContextInfSupBase and
  SetIntervalContextInfSup are two tightest
  implementations of set interval operations and
  functions (P1788 Level 2, InfSup_F).

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Factory classes for interval contexts

• Factory classes create particular instances of
  interval contexts
• SetIntervalContexts
• KaucherIntervalContexts

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Exact context

• Static method getExact() creates the exact
  context
• All operations in the exact context return
  intervals with rational bounds P1788 Level 1
  results or throw IrrationalException

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InfSup_F contexts

• Static method getInfSup(BinaryValueSet
  numberFormat) creates the InfsSup_F contexts with
  binary floating-point interval representations
  (BINARY32, BINARY64, BINARY128, , BINARY1024)

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Core Module Dependencies Graph
External dependencies
JInterval packages
jintervalAggregator of JInterval
jinterval-rational-java Rational numbers
boehm-crealsBoehms constructive reals
jinterval-interval-java Intervals, IAs, interval
elem. functions
fortress-roundingRounding class from Fortress
commons-math3Apache Commons Math 3.0
jinterval-ilsInterval linear equation system
solver
lpsolveJava port of lp_solve
jinterval-irInterval regression solver
mpfr-adapterJNA adapter for native GNU MPFR
jnaJava access to native libraries
large-test-javaJInterval tests
commons-compressApache Commons Compress 1.4
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JInterval Library Principles, Development, and
Perspectives
FUNCTIONALITY AND EXAMPLES
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Functionality of JInterval (Youth)

• Rational arithmetic
• flexible inner representation (rational,
  binary32, binary64, binary128,)
• exact and approximate operations
• Extended Rational arithmetic
• Rational -8 , 8
• Interval Arithmetic
• Set-based
• Kaucher
• Elementary Functions
• According to P1788

• Dense Vectors and Matrices
• Rational, extended rational
• Interval
• Solvers
• ILS Solvers
• Hansen-Bliek-Rohn-Ning-Kearfott enclosure
  Gauss-Seidel
• Subdifferential Newton
• Interval linear regression solver
• Data consistency check
• Outlier detection
• Object status detection
• Interval prediction

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Example 1.1. Contexts and Simple Expressions
, ,
, .
xy 3.0,5.0 x/y 0.333251953125,1.0
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Example 1.2. Contexts and Simple Expressions
, ,
, .
xy 3.0,5.0 x/y 0.3333333134651184,1.0
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Example 1.3. Contexts and Simple Expressions
, ,
, .
xy 3.0,5.0 x/y 0x155555555555555555555555
55555p-114,0x1p0 (0.3333333333333333,1.0
)
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Example 1.4. Contexts and Simple Expressions
, ,
, .
xy 3.0,5.0 x/y 0x1/0x320,0x1p0(0.333
3333333333333,1.0)
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Example 2.1. Decorations
sqrt(0.0,1.0) 0.0,1.0 COM sqrt(-1.0,1.0)
0.0,1.0 CON sqrt(-2.0,-1.0) EMPTY
NDF sqrt(EMPTY) EMPTY SAF
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Example 2.2. Decorations
1/0.0,0.0 EMPTY NDF 1/0.0,1.0
1.0,Infinity CON 1/4.9E-324,1.0
1.0,Infinity SAF x 0.0,Infinity SAF y
1/x 0.0,Infinity CON
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Example 3. (Rump)
Compute for
, and
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Example 3. (Rump) using ExtendedRational
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Example 3. (Rump) using ExtendedRational
BINARY16 rNaN BINARY32
r1.172603964805603 0x9617e3p-23
BINARY64 r-1.1805916207174113E21 -0x1p70
BINARY128 r1.1726039400531787
0x12c2fc595b06beb74a518f018c093p-112
BINARY256 r-0.8273960599468214
-0x69e81d3527ca0a45ad7387f39fb6bbbee6d0899f57af4ec
62443141c771p-235 Exact r-0.8273960599468
214 -0xd5ef/0x10292-4
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Example 3. (Rump) using SetInterval
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Example 3. (Rump) using SetInterval
BINARY16 i -Infinity,Infinity
BINARY32 i -6.972078301255262E30,6.97207890
5718172E30 BINARY64 i
-8.264141345021879E21,5.902958103587058E21
BINARY128 i -0xffb4f40e9a93e50522d6b9c3f9dp
-98,0x12c2fc595b06beb74a518f018c093p-112
(-1022.8273960599469,1.1726039400531787)
BINARY256 i -0x69e81d3527ca0a45ad7387f39fb6
bbbee6d0899f57af4ec62443141c771p-235,
-0xd3d03a6a4f94148b5ae70fe73f6d777dcda1133eaf5e9d8
c48862838ee1p-236 (-0.8273960599468214,-0.82
73960599468213) Exact i
-0xd5ef/0x10292-4,-0xd5ef/0x10292-4
(-0.8273960599468214,-0.8273960599468213)
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Example 4. HBRNK enclosure
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Example 4. HBRNK enclosure using MatlabOps
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Example 5. Gauss-Seidel Solver
A / 2.0,4.0 -2.0,0.0 -1.0,0.0
2.0,4.0 \ b / 1.0,2.0 -2.0,2.0
\ x / -1.0,4.0 -1.5,3.0 \
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JInterval Library Principles, Development, and
Perspectives
APPLICATIONS
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P1788 Test framework
TestSet3.dat
IntervalLibraries
Adapters
TestSet2.dat
Boost
testBoost
TestSet1.dat
CXSC
testCXSC
Launcher
testFilib
Filib
Report
MPFI
testMPFI
testPROFIL
PROFIL
ltNONAMEgt
testltNONAMEgt
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P1788 Test framework
IntervalLibraries
Adapters
TestSet3.dat
Boost
testBoost
TestSet2.dat
CXSC
testCXSC
Report
Launcher
testFilib
Filib
MPFI
testMPFI
TestSet1.dat
testPROFIL
PROFIL
ltNONAMEgt
testltNONAMEgt
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P1788 Test framework
IntervalLibraries
Adapters
Boost
testBoost
TestSet3.dat
CXSC
testCXSC
Report
Launcher
testFilib
Filib
MPFI
testMPFI
TestSet2.dat
testPROFIL
PROFIL
ltNONAMEgt
testltNONAMEgt
TestSet1.dat
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P1788 Test framework. Sample Run
testDemo.dat
div1,2 0,11,2 0,0
sqrt-Infinity,0-Infinity,Infinity
pown0,0 0
Report
Filib 3.0.2 div 1.0,2.0 0.0,1.0
1.0,Infinity 1.0,Infinity Ok div 1.0,2.0
0.0,0.0 EMPTY 1.7976931348623157E308,Infi
nity NOT TIGHT! sqrt -Infinity,0.0 0.0,0.0
0.0,0.0 Ok sqrt -Infinity,Infinity
0.0,Infinity -4.9E-324,Infinity NOT
TIGHT! pown 0.0,0.0 0 1.0,1.0 1.0,1.0
Ok Boost 1.48.0 div 1.0,2.0 0.0,1.0
1.0,Infinity 1.0,Infinity Ok div 1.0,2.0
0.0,0.0 EMPTY EMPTY Ok sqrt
-Infinity,0.0 0.0,0.0 0.0,0.0 Ok sqrt
-Infinity,Infinity 0.0,Infinity
0.0,Infinity Ok pown 0.0,0.0 0 1.0,1.0
EMPTY CONTAINMENT FAILURE!!! MPFI
1.5.1 div 1.0,2.0 0.0,1.0 1.0,Infinity
1.0,Infinity Ok div 1.0,2.0 0.0,0.0
EMPTY -Infinity,Infinity NOT TIGHT! sqrt
-Infinity,0.0 0.0,0.0 EMPTY CONTAINMENT
FAILURE!!! sqrt -Infinity,Infinity
0.0,Infinity EMPTY CONTAINMENT
FAILURE!!! Library has no Operation "pown" in
line 7 pown
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KNIME

• KNIME open source data mining platform

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KNIME Interval Tools

• KNIME open source data mining platform

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KNIME Nodes for Interval Regression
Interval Regression( Learner)
Interval Regression( Predictor)
Calculates interval prediction Y for X using
model Y F(X, A)
Builds linear interval regression model Y
f(X, A)
IR Outlier Detector
IR Consistency
Detects outliers (observations with
underestimated error bound)
Checks consistency of input data and sets flow
control variable for IF switch
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KNIME Nodes for ILS Solving

• In previous version of Interval Tools

Computes inner/outer estimate of united/tolerable
solution set and visualizes 2D/3D united
solution set
ILS Solver
Kraemer W. Computing and visualizing solutions
sets of interval linear systems, Serdica J.
Computing 1(4) 2007, 455-468.
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KNIME Nodes for ILS Solving
ILS Solver
ILS United Solution Set View
Visualizes 2D or 3D united solution set
Computes outer and inner estimates for united
solution set of ILS Axb

• Methods for united solution set estimation
• Outer Hansen-Bliek-Rohn-Ning-Kearfott enclosure
  Gauss-Seidel
• Inner Subdifferential Newton

• Method
• I. A. Sharayas algorithm for visualization of
  AE-solution sets

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KNIME Workflow for Image Recognition
Get A1, b
Solve A1xb
Metrics values
Read reference images
Join
Convert to matrices
Solve A2xb
Join
Build interval matrices
Get A2, b
Join
Calc metrics
Read imageto recognize
Convert to matrices
Solve non-interval ILS
Metrics histogram
Prolubnikov A.V., Silitskiy S.A. On solving the
problem of numeric matrices recognition using
estimates of solution sets of interval systems of
equations // Comp. Math. Proceedings of XIV
Baikal International School-Seminar Methods of
optimization and its applications
Irkutsk-Baikal, July 2-8 ???? 2008. Vol. 3.
Irkutsk ISEM SB RAS, 2008. pp. 152-157. (in
Russian)
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KNIME Workflow for Image Recognition
Get A1, b
Solve A1xb
Metrics values
Read reference images
Join
Convert to matrices
Solve A2xb
Join
Build interval matrices
Get A2, b
Join
Calc metrics
Read imageto recognize
Convert to matrices
Solve non-interval ILS
Metrics histogram
Prolubnikov A.V., Silitskiy S.A. On solving the
problem of numeric matrices recognition using
estimates of solution sets of interval systems of
equations // Comp. Math. Proceedings of XIV
Baikal International School-Seminar Methods of
optimization and its applications
Irkutsk-Baikal, July 2-8 ???? 2008. Vol. 3.
Irkutsk ISEM SB RAS, 2008. pp. 152-157. (in
Russian)
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Mobile Application Affiche

• Modeling positional uncertainties for GPSGSM
  navigation using circular complex arithmetic

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JInterval Library Principles, Development, and
Perspectives
PERSPECTIVES
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Future Directions

• Holding P1788 compliance of JInterval
• Developing optional plugins for
  platform-dependent effective implementations of
  inner layers through JNI
• MPFR for fast multiple precision floating-point
  arithmetic
• BLAS for fast linear algebra operations
• Developing API for access to JInterval from
  other programming languages
• Scala
• High-level functionality replenishment

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High-Level Functionality Replenishment

• Enhancing MatlabOps
• ILS tolerable solution set estimators
• ILS AE-solution set visualization (I. Sharaya)
• Global Optimization Solver (N. Panov S. Shary)
• Randomized Branch Bound
• Interval simulating annealing
• Interval Genetic algorithm
• Multi-method algorithm
• ODE Solver (D. Nadezhin)

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How to Get and Contribute to JInterval?

• java.net/projects/jinterval
• Source codes (SVN)
• JavaDoc
• Wiki
• Tutorial
• Developers forum
• Mail lists

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Acknowledgements

• Maksim V. Danilov
• Kirill S. Dronov
• Walter Krämer
• Nikita V. Panov
• Gregor Paw
• Anton E. Sartakov
• Andrey S. Samoilov
• Sergey P. Shary
• Irina A. Sharaya
• Egor N. Tepikin
• Leo N. Tolstoy
• IEEE Interval Standard Working Group P1788

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