jinterval library : p rinciples , development , and p erspectives

JInterval Library: Principles, Development, and Perspectives

Sergei I. ZhilinAltai State UniversityBarnaul, [email protected]

Dmitry Ju. NadezhinOracle LabsZelenograd, [email protected]

15’th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerical Computations

Novosibirsk, September 23–29, 2012SCAN’2012

Outline

Why Interval Computations in Java Virtual Machine (JVM)? JInterval Evolution Architecture Functionality and Examples Applications Perspectives

Novosibirsk, 26 Sep 2012 SCAN 2012 2 / 60

WHY INTERVAL COMPUTATIONS IN JVM?JInterval Library: Principles, Development, and Perspectives


Java Is Popular

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Java Is Popular

RedMonk’s language ranking for September 2012

Novosibirsk, 26 Sep 2012 SCAN 2012

Popularity Rank on Github.com (by # of projects)

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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


Interval Computations in Java

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, С/С++


Is Java suitable for scientific computing?

*Kahan W., Darcy J.D. How Java’s Floating-Point Hurts Everyone Everywhere//ACM 1998 Workshop on Java forHigh–Performance 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


JINTERVAL EVOLUTIONJInterval Library: Principles, Development, and Perspectives


Stages of JInterval Evolution

Sep 2008 JInterval is started as undergraduate student project ”Childhood” at Altai State University (Barnaul, Russia)

http://code.google.com/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 orhttp://jinterval.java.net


JInterval (Boyhood): A Priori Requirements

The library

1. Must be clear and easy to use

2. Should provide flexibility in the choice of interval algebra for computations

3. Should provide flexibility to extend its functionality

4. Should provide flexibility in choosing precision of interval boundaries and associated rounding policies

5. Must be portable

6. Should provide high performance7. Must be open source


Prio

rity

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


JInterval (Boyhood): Type Hierarchy


Interval

ClassicRealInterval ComplexInterval

RealInterval

DoubleIntervalDoubleInterval

RationalIntervalRationalInterval

ComplexIntervalCircleComplexIntervalCircle

ComplexIntervalRectangleComplexIntervalRectangle

ComplexIntervalPolarComplexIntervalPolar

ComplexIntervalRingComplexIntervalRing

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 + y*z

ARCHITECTUREJInterval Library: Principles, Development, and Perspectives


Class Diagram (package net.java.jinterval.interval)


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’

– KaucherIntervalExtends Interval with methods for

flavor ‘Kaucher interval’

– ClassicInterval Extends all flavors, because can be

mapped to related flavor intervals

Interface Interval: common methods of all flavors


Interval defines common methods of all interval flavors

Defines numerical and boolean operations only

IntervalContext: interval operations interface


Generic interface IntervalContext defines signature for interval-valued methods

KaucherIntervalContext: interval flavor interface


KaucherIntervalContext extends IntervalContextand binds type variable I to Kaucher interval flavor

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).


Factory classes for interval contexts


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

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

InfSup_F contexts


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

jintervalAggregator of JInterval

Core Module Dependencies Graph


jinterval-irInterval regression solverjinterval-irInterval regression solver

jinterval-ilsInterval linear equation system solverjinterval-ilsInterval linear equation system solver

jinterval-interval-java Intervals, IAs, interval elem. functionsjinterval-interval-java Intervals, IAs, interval elem. functions

jinterval-rational-java Rational numbersjinterval-rational-java Rational numbers

fortress-roundingRounding class from Fortressfortress-roundingRounding class from Fortress

boehm-crealsBoehm’s constructive realsboehm-crealsBoehm’s constructive reals

mpfr-adapterJNA adapter for native GNU MPFRmpfr-adapterJNA adapter for native GNU MPFR

commons-math3Apache Commons Math 3.0commons-math3Apache Commons Math 3.0

lpsolveJava port of lp_solvelpsolveJava port of lp_solve

jnaJava access to native librariesjnaJava access to native libraries

large-test-javaJInterval testslarge-test-javaJInterval tests

commons-compressApache Commons Compress 1.4commons-compressApache Commons Compress 1.4

External dependenciesExternal dependenciesJInterval packagesJInterval packages

FUNCTIONALITY AND EXAMPLESJInterval Library: Principles, Development, and Perspectives


Functionality of JInterval (Youth)

Rational arithmetic– flexible inner representation

(rational, binary32, binary64, binary128,…)

– exact and approximate operations

Extended Rational arithmetic– Rational + {-∞ , +∞}

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


Example 1.1. Contexts and Simple Expressions


x+y = [3.0,5.0]x/y = [0.333251953125,1.0]x+y = [3.0,5.0]x/y = [0.333251953125,1.0]

, , , .



x+y = [3.0,5.0]x/y = [0.3333333134651184,1.0]x+y = [3.0,5.0]x/y = [0.3333333134651184,1.0]

, , , .



x+y = [3.0,5.0]x/y = [+0x15555555555555555555555555555p-114,+0x1p0] ([0.3333333333333333,1.0])

x+y = [3.0,5.0]x/y = [+0x15555555555555555555555555555p-114,+0x1p0] ([0.3333333333333333,1.0])

, , , .



x+y = [3.0,5.0]x/y = [+0x1/0x3*2^0,+0x1p0]([0.3333333333333333,1.0])x+y = [3.0,5.0]x/y = [+0x1/0x3*2^0,+0x1p0]([0.3333333333333333,1.0])

, , , .

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

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

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

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

Example 3. (Rump)


Compute for , and

Example 3. (Rump) using ExtendedRational


Example 3. (Rump) using ExtendedRational


=== BINARY16 ===r=NaN=== BINARY32 ===r=1.172603964805603 +0x9617e3p-23=== BINARY64 ===r=-1.1805916207174113E21 -0x1p70=== BINARY128 ===r=1.1726039400531787 +0x12c2fc595b06beb74a518f018c093p-112=== BINARY256 ===r=-0.8273960599468214 -0x69e81d3527ca0a45ad7387f39fb6bbbee6d0899f57af4ec62443141c771p-235=== Exact ===r=-0.8273960599468214 -0xd5ef/0x1029*2^-4

=== BINARY16 ===r=NaN=== BINARY32 ===r=1.172603964805603 +0x9617e3p-23=== BINARY64 ===r=-1.1805916207174113E21 -0x1p70=== BINARY128 ===r=1.1726039400531787 +0x12c2fc595b06beb74a518f018c093p-112=== BINARY256 ===r=-0.8273960599468214 -0x69e81d3527ca0a45ad7387f39fb6bbbee6d0899f57af4ec62443141c771p-235=== Exact ===r=-0.8273960599468214 -0xd5ef/0x1029*2^-4

Example 3. (Rump) using SetInterval


Example 3. (Rump) using SetInterval


=== BINARY16 ===i = [-Infinity,Infinity]=== BINARY32 ===i = [-6.972078301255262E30,6.972078905718172E30]=== BINARY64 ===i = [-8.264141345021879E21,5.902958103587058E21]=== BINARY128 ===i = [-0xffb4f40e9a93e50522d6b9c3f9dp-98,+0x12c2fc595b06beb74a518f018c093p-112] ([-1022.8273960599469,1.1726039400531787])=== BINARY256 ===i = [-0x69e81d3527ca0a45ad7387f39fb6bbbee6d0899f57af4ec62443141c771p-235, -0xd3d03a6a4f94148b5ae70fe73f6d777dcda1133eaf5e9d8c48862838ee1p-236] ([-0.8273960599468214,-0.8273960599468213])=== Exact ===i = [-0xd5ef/0x1029*2^-4,-0xd5ef/0x1029*2^-4] ([-0.8273960599468214,-0.8273960599468213])

Example 4. HBRNK enclosure


Example 4. HBRNK enclosure using MatlabOps


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] |\

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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APPLICATIONSJInterval Library: Principles, Development, and Perspectives


P1788 Test framework


FilibFilib

CXSCCXSC

PROFILPROFIL

BoostBoost

MPFIMPFILauncherLauncher

<NONAME><NONAME>

TestSet2.datTestSet2.dat


ReporttestFilibtestFilib

testCXSCtestCXSC

testPROFILtestPROFIL

testBoosttestBoost

testMPFItestMPFI

test<NONAME>test<NONAME>

TestSet3.datTestSet3.datIntervalLibrariesIntervalLibraries AdaptersAdapters



FilibFilib

CXSCCXSC

PROFILPROFIL

BoostBoost


<NONAME><NONAME>





testCXSCtestCXSC


testBoosttestBoost

testMPFItestMPFI


IntervalLibrariesIntervalLibraries AdaptersAdapters

* div[1,2] [0,1][1,2] [0,0]* sqrt[-Infinity,0][-Infinity,Infinity]* pown[0,0] 0

* div[1,2] [0,1][1,2] [0,0]* sqrt[-Infinity,0][-Infinity,Infinity]* pown[0,0] 0

P1788 Test framework. Sample RuntestDemo.dattestDemo.dat

== Filib 3.0.2div [1.0,2.0] [0.0,1.0] = [1.0,Infinity] : [1.0,Infinity] Okdiv [1.0,2.0] [0.0,0.0] = [EMPTY] : [1.7976931348623157E308,Infinity] NOT TIGHT!sqrt [-Infinity,0.0] = [0.0,0.0] : [0.0,0.0] Oksqrt [-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 1.48.0div [1.0,2.0] [0.0,1.0] = [1.0,Infinity] : [1.0,Infinity] Okdiv [1.0,2.0] [0.0,0.0] = [EMPTY] : [EMPTY] Oksqrt [-Infinity,0.0] = [0.0,0.0] : [0.0,0.0] Oksqrt [-Infinity,Infinity] = [0.0,Infinity] : [0.0,Infinity] Okpown [0.0,0.0] 0 = [1.0,1.0] : [EMPTY] CONTAINMENT FAILURE!!!==

== MPFI 1.5.1div [1.0,2.0] [0.0,1.0] = [1.0,Infinity] : [1.0,Infinity] Okdiv [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==

== Filib 3.0.2div [1.0,2.0] [0.0,1.0] = [1.0,Infinity] : [1.0,Infinity] Okdiv [1.0,2.0] [0.0,0.0] = [EMPTY] : [1.7976931348623157E308,Infinity] NOT TIGHT!sqrt [-Infinity,0.0] = [0.0,0.0] : [0.0,0.0] Oksqrt [-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 1.48.0div [1.0,2.0] [0.0,1.0] = [1.0,Infinity] : [1.0,Infinity] Okdiv [1.0,2.0] [0.0,0.0] = [EMPTY] : [EMPTY] Oksqrt [-Infinity,0.0] = [0.0,0.0] : [0.0,0.0] Oksqrt [-Infinity,Infinity] = [0.0,Infinity] : [0.0,Infinity] Okpown [0.0,0.0] 0 = [1.0,1.0] : [EMPTY] CONTAINMENT FAILURE!!!==

== MPFI 1.5.1div [1.0,2.0] [0.0,1.0] = [1.0,Infinity] : [1.0,Infinity] Okdiv [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==

ReportReport


KNIME

KNIME— open source data mining platform


KNIME Interval Tools

KNIME— open source data mining platform


KNIME Nodes for Interval Regression


Builds linear interval regression model Y = f(X, A)

Interval Regression( Learner)

IR Outlier Detector

Calculates interval prediction Y* for X* using model Y = F(X, A)

Detects outliers (observations with underestimated error bound)

Interval Regression( Predictor)

IR Consistency

Checks consistency of input data and sets flow control variable

for IF switch

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

*Kraemer W. Computing and visualizing solutions sets of interval linear systems, Serdica J. Computing 1(4) 2007, 455-468.

ILS Solver

KNIME Nodes for ILS Solving


Methods for united solution set estimation: – Outer:

Hansen-Bliek-Rohn-Ning-Kearfott enclosure + Gauss-Seidel

– Inner: Subdifferential Newton

Computes outer and inner estimates for united solution set of ILS Ax=b

ILS Solver ILS United Solution Set View

Visualizes 2D or 3D united solution set

Method: – I. A. Sharaya’s

algorithm for visualization of AE-solution sets

Under construction

KNIME Workflow for Image Recognition


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 июля 2008. Vol. 3. – Irkutsk: ISEM SB RAS, 2008. – pp. 152-157. (in Russian)

Read reference images

Read imageto recognize

Convert to matrices

Convert to matrices

Join Build interval matrices

Get A1, b

Get A2, b

Solve A1*x=b

Solve A2*x=b

Solve non-interval ILS

Join

JoinCalc metrics

Metrics values

Metrics histogram

Mobile Application “Affiche”

Modeling positional uncertainties for GPS+GSM navigation using circular complex arithmetic


PERSPECTIVESJInterval Library: Principles, Development, and Perspectives


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


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)


How to Get and Contribute to JInterval?

java.net/projects/jintervalSource codes (SVN)JavaDocWikiTutorialDevelopers’ forumMail lists


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


jinterval library : p rinciples , development , and p erspectives

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