NSWC Mathematical Library Last Changed : 11 June 1993 Computing and Network Services, University of Alberta Technical Contacts: Ron Torgerson: X9351, torg@norge.ucs.ualberta.ca Eva Wong: X2047, ewong@seymour.ucs.ualberta.ca Introduction The NSWC Mathematics Subroutine Library is a collection of Fortran 77 routines specializing in numerical mathematics collected and developed by the: Computing Systems and Networks Division Strategic and Space Systems Department U.S. Naval Surface Warfare Center Dahlgren Division Dahlgren Virginia 22448-5000 This software is made available, without cost, to the general scientific community. The 1993 edition is an update of the 1990 edition. NSWC has made every effort to include only reliable, transportable, reasonably efficient and easy to use code in this library. They have thoroughly tested all the routines on a variety of machines ranging from supercomputers to PC's. Program Location A compressed version of the complete Fortran source of this library is stored in the file nswc.f.Z on the CNS anonymous file server. To get this source file and a copy of this introduction, do the following: %ftp ftp.srv.ucs.ualberta.ca Name (ftp.srv.ualberta.ca:yourid): anonymous Password: ftp> cd pub/unix/numerical/nswc ftp> get nswc.f.Z ftp> get Intro ftp> quit %uncompress nswc.f.Z Machine Dependence The authors have attempted to write code that will work on virtually any computer. All the machine dependences are dealt with in the first routine INTEGER FUNCTION IMPAR which sets the machine constants such as the number of bits in an integer, the base of the floating point arithmetic and the mantissa size for single and double precision floating point numbers. The source contains many commented out possibilities. The ones that were not correspond to the IBM PC. Since these agree with the IEEE arithmetic standard and almost all modern Micros and workstations represent their numbers in the IEEE fashion most people won't need to change anything. This is valid for Mac, PC, SUN, IBM RS6000, HP, SGI ... If there is any doubt, or if your machine is not listed in IMPAR, the second program SUBROUTINE MACH can be used to help determine the machine constants of your computer. Documentation The 610 page user manual NSWC LIBRARY OF MATHEMATICS SUBROUTINES gives a detailed description of the 576 user level routines. A copy is available from the help desk in room 302 General Services Building. It appears that there is no postscript file version; however the manual is not copyrighted so it or parts of it may be reproduced freely. The manual may be borrowed for this purpose. The various chapters of the library are: Elementary Operations Geometry Special Functions Polynominals Solutions of Nonlinear Equations Vectors Matrices Large Dense Systems of Linear Equations Banded Matrices Sparse Matrices Eigenvalues and Eigenvectors L1 Solution of Linear Equations Least Squares Solutions of Linear Equations Optimization Transforms Approximation of Functions Curve Fitting Surface Fitting over Rectangular Grids Surface Fitting over Arbitrarly Positioned data Points Manifold Fitting Numerical Integration Integral Equations Ordinary Differential Equations/Initial Value Problems Partial Differential Equations Discrete Random Number Generation Continuous Random Number Generation In order to determine if one or more NSWC routines can serve your purposes please refer to the manual's table of contents reproduced below. In what follows, the vertical bar indicates new routines. The numbers at the right refer to page numbers of the user manual. NSWC LIBRARY TABLE OF CONTENTS Elementary Operations |Machine Constants Q SPMPAR, DPMPAR, IPPMAR .................3 |Argument Bounds for the Exponential Function - | EPSLN, EXPARG, DEPSLN, DXPARG..........................5 |Sorting Lists Q ISHELL, SHELL, AORD, RISORT, SHELL2, DSORT, | DAORD, DISORT, DDSORT, QSORTI, QSORTR, QSORTD, IORDER, | RORDER, DORDER ........................................7 |Cube Root - CBRT, DCBRT ...................................11 Four Quadrant Arctangent - ARTNQ, DARTNQ...................11 Length of a Two-Dimensional Vector - CPABS, DCPABS ........11 Reciprocal of a Complex Number - CREC, DCREC ..............13 |Division of a Complex Number - CDVI,DIVID..................13 Square Root of a Double Precision Complex Number - DCSQR...13 Conversion of Polar to Cartesian Coordinates Q POCA .......15 Conversion of Cartesian to Polar Coordinates - CAPO .......15 Rotation of Axes - ROTA ...................................15 Planar Givens Rotations - SROTG, DROTG ....................17 Three Dimension Rotations - ROT3 ..........................19 Rotation of a Point on the Unit Sphere to the North Pole - CONSTR ...............................................21 |Computation of the Angle Between Two Vectors - ANG ........23 |Trigonometric Functions - SIN1, COS1, DSIN1, DCOS1 ........25 |Hyperbolic Sine and Cosine Functions SNHCSH ...............27 |Exponentials Q REXP, DREXP ................................29 Logarithms - ALNREL, RLOG, RLOG1, DLNREL, DRLOG, DRLOG1 ...31 Geometry Determining if a Point is Inside or Outside a Polygon - LOCPT ................................................33 |Intersection of a Straight Line and Polygonal Path - PFIND.35 The Convex Hull for a Finite Planar Set Q HULL ............37 Areas of Planar Polygons - PAREA ..........................39 Hamiltonian Circuits - HC .................................41 Special Functions Error Function - CERF, CERFC, ERF, ERFC, ERFC1, DCERF, DCERFC, DERF, DERFC, DERFC1 ...........................45 |Inverse Error Function - ERFI, DERFI ......................51 |Difference of Error Function - AERF, DAERF ................53 Normal Probability Distribution Function - PNDF ...........55 |Inverse Normal Probability Distribution Function - PNI,DPNI ..............................................57 |Dawson's Integral - DAW, DPDAW ............................59 Complex Fresnel Integral - CFRNLI .........................61 Real Fresnel Integrals - FRNL .............................63 Exponential Integral Function - CEXPLI, EXPLI, DEI, DEI1 ..65 Sine and Cosine Integral Functions - SI, CIN ..............69 |Exponential Exponential Integral Function - CEXEXI ........71 Dilogarithm Function - CLI, ALI ...........................73 Gamma Function - CGAMMA, GAMMA, GAMLN, DCGAMA, DGAMMA, DGAMLN .......................................75 Digamma Function - CPSI, PSI, DCPSI, DPSI .................79 |Derivatives of the Digamma Function - PSIDF ...............81 |Incomplete Gamma Ratio Functions - GRATIO, RCOMP, DGRAT, DRCOMP ...............................................83 |Inverse Incomplete Gamma Ratio Function - GAMINV, DGINV ...85 Logarithm of the Beta Function Q BETALN, DBETLN ...........87 Incomplete Beta Function - BRATIO, ISUBX, BRCOMP ..........89 Bessel Function Jv(z) - CBSSLJ,BSSLJ, BESJ ................91 Bessel Function Yv(z) - BSSLY .............................93 |Modified Bessel Function Iv(Z) - CBSSLI, BSSLI, BESI ......95 |Modified Bessel Function Kv(z) - CBESK, CBSSLK, BSSLK .....97 Airy Functions - CAI, CBI, AI, AIE, BI, BIE ...............99 Complete Complex Elliptic Integrals of the First and Second Kinds - CK, CKE ..............................103 Real Elliptic Integrals of the First and Second Kinds - ELLPI, RFVAL, RDVAL, DELLPI, DRFVAL, DRDVAL .........107 Real Elliptic Integrals of the Third Kind - EPI, RJVAL, DEPI, DRJVAL ............................111 Jacobian Elliptic Functions - ELLPF, ELPFC1 ..............115 Weierstrass Elliptic Function for the Equianharmonic and Lemniscatic Cases - PEQ, PEQ1, PLEM, PLEM1 ......119 Integral of the Bivariate Density Function over Arbitrary Polygons and Semi-infinite Angular Regions - VALR2 ..123 |Circular Coverage Function - CIRCV .......................125 |Elliptical Coverage Function Q PKILL .....................127 Polynomials Copying Polynomials - PLCOPY, DPCOPY .....................129 Addition of Polynomials - PADD, DPADD ....................131 Subtraction of Polynomials - PSUBT, DPSUBST ..............133 Multiplication of Polynomials - PMULT, DPMULT ............135 Division of Polynomials Q PDIV, DPDIV ....................137 Real Powers of Polynomials - PLPWR, DPLPWR ...............139 Inverses of Power Series - PINV, DPINV ...................141 Derivatives and Integrals of Polynomials - MPLNMV ........143 Evaluation of Chebyshev Expansions - CSEVL, DCSEVL .......145 Lagrange Polynomials Q LGRNGN, LGRNGV, LRGNGX ............147 Orthogonal Polynomials on Finite Sets - ORTHOS, ORTHOV, ORTHOX ..............................................149 Solutions of Nonlinear Equations |Zeros of Continuous Functions - ZEROIN, DZERO ............151 Solution of Systems of Nonlinear Equations - HBRD ........153 Solutions of Quadratic, Cubic, and Quartic Equations - QDCRT, CBCRT, QTCRT, DQDCRT, DCBCRT, DQTCRT ........155 Double Precision Roots of Polynomials - DRPOLY, DCPOLY ...157 |Accuracy of the Roots of Polynomial - RBND, CBND .........159 Vectors Copying Vectors Q SCOPY, DCOPY, CCOPY ....................161 Interchanging Vectors - SSWAP, DSWAP, CSWAP ..............163 Planar Rotation of Vectors - SROT, DROT, CSROT ...........165 |Modified Givens Rotations - SROTMG, DROTMG, SROTM, DROTM .167 Dot Products of Vectors - SDOT, DDOT, CDOTC, CDOTU .......171 Scaling Vectors - SSCAL, DSCAL, CSCAL, CSSCAL ............173 Vector Addition - SAXPY, DAXPY, CAXPY ....................175 Ll Norm of a Vector - SASUM, DASUM, SCASUM ...............177 L2 Norm of a Vector Q SNRM2, DNRM2, SCNRM2 ...............179 L0 Norm of a Vector - ISAMAX, IDAMAX, ICAMAX .............181 Matrices Packing and Unpacking Symmetric Matrices - MCVFS, DMCVFS, MCVSF, DMCVSF .......................................183 Conversion of Real Matrices to and from Double Precision Form - MCVRD, MDCVDR ................................185 Storage of Real Matrices in the Complex Matrix Format - MCVRC ...............................................187 The Real and Imaginary Parts of a Complex Matrix - CMREAL, CMIMAG.......................................189 Copying matrices - MCOPY, SMCOPY, DMCOPY, CMCOPY .........191 Computation of the Conjugate of a Complex Matrix - CMCONJ.193 Transposing Matrices Q TPOSE, DTPOSE, CTPOSE, TIP, DTIP, CTIP ..........................................195 Computing Adjoints of Complex Matrices - CMADJ, CTRANS ...197 Matrix Addition - MADD, SMADD, DMADD, CMADD ..............199 Matrix Subtraction - MSUBT, SMSUBT, DMSUBT, CMSUBT .......201 Matrix Multiplication - MTMS, DMTMS, CMTMS, MPROD, DMPROD, CMPROD ......................................203 Product of a Packed Symmetric Matrix and a Vector - SVPRD, DSVPRD .......................................205 Transpose Matrix Products - TMPROD .......................207 Symmetric Matrix Products - SMPROD .......................209 Kronecker Product of Matrices - KPROD, DKPROD, CKPROD ....211 |Rank of a Real Matrix - RNK, DRNK ........................213 |Inverting General Real Matrices and Solving General | Systems of Real Linear Equations - CROUT,KROUT, | NPIVOT, MSLV, DMSLV, MSLV1, DMSLV1 ............... 215 Solutions of Real Equations with Iterative Improvement - SLVMP ...............................................221 Solutions of Almost Block Diagonal Systems of Linear Equations - ARCECO, ARCESL ..........................223 Solution of Almost Block Tridiagonal Systems of Linear Equations Q BTSLV ...................................225 Inverting Symmetric Real Matrices and Solving Symmetric Systems of Real Linear Equations - SMSLV, DSMSLV ....227 Inverting Positive Definite Symmetric Matrices and Solving Positive Definite Symmetric Systems of Linear Equations - PCHOL,DPCHOL .....................231 Solution of Toeplitz Systems of Linear Equations - TOPLX, DTOPLX .......................................233 Inverting General Complex Matrices and Solving General Systems of Complex Linear Equations - CMSLV, CMSLV1, DCMSLV ...............................235 Solution of Complex Equations with Iterative Improvement - CSLVMP ..............................................239 Singular Value Decomposition of a Matrix - SSVDC,DSVDC, CSVDC ...............................................241 Evaluation of the Characteristic Polynomial of a Matrix - DET, DPDET, CDET ...........................243 Solution of the Matrix Equation AX + XB = C - ABSLV, DABSLV .......................................245 Solution of the Matrix Equation AtX + XA = C where C is Symmetric - TASLV, DTASLV ...........................247 Solution of the Matrix Equation - AX2 + BX + C = O - SQUINT ..............................................249 Exponential of a Real Matrix - MEXP, DMEXP ...............251 Large Dense Systems of Linear Equations Solving systems of 200-400 Linear Equations - LE, DPLE, CLE .......................................253 Banded Matrices Band Matrix Storage ......................................255 |Conversion of Banded Matrices to and from the | Standard Format - CVBR, CVBD, CVBC, CVRB, | CVDB, CVCB, CVRB1,CVDB1, CVCB1 ......................257 |Conversion of Banded Matrices to and from Sparse Form - | MCVBS, DMCVBS, CMCVBS, MCVSB, DMCVSB, CMCVSB ....... 259 |Conversion of Banded Real Matrices to and from | Double Precision Form - BCVRD, BCVDR ............... 261 |The Real and Imaginary Parts of a Banded | Complex Matrix - BREAL, BIMAG .................... 263 |Computing A + Bi for Banded Real Matrices A and B - BCVR..265 |Transposing Banded Matrices Q BPOSE, DBPOSE, CBPOSE ......267 |Addition of Banded Matrices - BADD, DBADD, CBADD .........269 |Subtraction of Banded Matrices - BSUBT, DBSUBT, CBSUBT ...271 |Multiplication of Banded Matrices - BPROD,DBPROD,CBPROD ..273 Product of a Real Banded Matrix and Vector - BVPRD, BVPRD1, BTPRD, BTPRD1 .................. 275 |Product of a Double Precision Banded Matrix and Vector - DBVPD, DBVPD1, DBTPD, DBTPD1 ........................277 Product of a Complex Banded Matrix and Vector - CBVPD, CBVPD1, CBTPD, CBTPD1 ..................... 279 |L1 Norm of a Real Banded Matrix - B1NRM, DB1NRM ..........281 |L0 Norm of a Real Banded Matrix - BNRM, DBNRM ............283 Solution of Banded Systems of Real Linear Equations - BSLV, BSLV1 .........................................285 |Computation of the Condition Number of a Real | Banded Matrix - B1CND ...............................287 |Double Precision Solution of Banded Systems of | Real Linear Equations - DBSLV, DBSLV1 ...............289 |Computation of the Condition Number of a Double Precision Banded Matrix - DB1CND .............291 Solution of Banded Systems of Complex Linear Equations - CBSLV, CBSLV1 .......................................293 Sparse Matrices Storage of Sparse Matrices ...............................295 Conversion of Sparse Matrices to and from the Standard Format - CVRS, CVDS, CVCS, CVSR, CVSD, CVSC .........297 Conversion of Spase Real Matrices to and from Double Precision Form - SCVRD, SCVDR ................299 The Real and Imaginary Parts of a Sparse Complex Matrix - CSREAL, CSIMAG ......................................301 Computing A + Bi for Sparse Real Matrices A and B Q SCVRC ...............................................303 Copying Sparse Matrices - RSCOPY, DSCOPY, CSCOPY ........305 Computing Conjugates of Sparse Complex Matrices - SCONJ ..307 Transposing Sparse Real Matrices - RPSOE, RPOSE1 .........309 Transposing Sparse Double Precision Matrices - DPOSE, DPOSE1 .......................................311 Transposing Sparse Complex Matrices - CPOSE, CPOSE1 ......313 Addition of Sparse Matrices - SADD, DSADD, CSADD .........315 Subtraction of Sparse Matrices Q SSUBT, DSSUBT, CSSUBT ...317 Multiplication of Sparse Matrices - SPROD,DSPROD,CSPROD ..319 Product of a Real Sparse Matrix and Vector - MVPRD, MVPRD1, MTPRD, MTPRD1 ........................321 Product of a Double Precision Sparse Matrix and Vector Q DVPRD, DVPRD1, DTPRD, DTPRD1 ........................323 Product of a Complex Sparse Matrix and Vector - CVPRD, CVPRD1, CTPRD, CTPRD1 ........................325 |L1 Norm of a Sparse Real Matrix - S1NRM, DS1NRM ..........327 |L0 Norm of a Sparse Real Matrix - SNRM, DSNRM ............329 Ordering the Rows of a Sparse Matrix by Increasing Length Q SPORD ...........................331 Reordering Sparse Matrix into Block Triangular Form Q BLKORD ..............................................333 Solution of Sparse Systems of Real Linear Equations - SPSLV, RSLV, TSLV ...................................335 |Computation of the Condition Number of a Real | Sparse Matrix - S1CND ...............................339 Double Precision Solution of Sparse Systems of Real Linear Equation - DSPSLV, DSLV, DTSLV ..........341 |Computation of the Condition Number of a | Double Precision Sparse Matrix - DS1CND .............345 Solution of Sparse Systems of Complex Linear Equations - CSPSLV, CSLV, CTSLV .................................347 Eigenvalues and Eigenvectors Computation of Eigenvalues of General Real Matrices - EIG, EIG1 ................................... .......351 Computation of Eigenvalues and Eigenvectors of General Real Matrices - EIGV, EIGV1 .................353 Double Precision Computation of Eigenvalues of Real Matrices - DEIG ................................355 Double Precision Computation of Eigenvalues and Eigenvectors of Real Matrices - DEIGV ...............357 Computation of Eigenvalues of Symmetric Real Matrices - SEIG, SEIG1 .........................................359 Computation of Eigenvalues and Eigenvectors of Symmetric Real Matrices - SEIGV, SEIGV1 .............361 |Double Precision Computation of Eigenvalues of | Symmetric Real Matrices - DSEIG .....................363 |Double Precision Computation of Eigenvalues and | Eigenvectors of Symmetric Real Matrices - DSEIGV ....365 Computation of Eigenvalues of Complex Matrices - CEIG ....367 Computation of Eigenvalues and Eigenvectors of Complex Matrices - CEIGV ............................369 Double Precision Computation of Eigenvalues of Complex Matrices - DCEIG ............................371 Double Precision Computation of Eigenvalues and Eigenvectors of Complex Matrices - DCEIGV ...........373 L1 Solution of Linear Equations L1 Solution of Systems of Linear Equations with Equality and Inequality Constraints - CL1 ....................375 Least Squares Solution of Linear Equations |Least Squares Solution of Systems of Linear Equations - | LLSQ, LSQR, HFTI, HFTI2 .......................... 377 Least Squares Solution of Overdetermined Systems of Linear Equations with Iterative Improvement - LLSQMP .......383 |Double Precision Least Squares Solution of Systems of | Linear Equations - DLLSQ, DLSQR, DHFTI, DHFTI2 .....385 Least Squares Solution of Systems of Linear Equations with Equality and Inequality Constraints - LSEI ..........391 Least Squares Solution of Systems of Linear Equations with Equality and Nonnegativity Constraints - WNNLS ......395 Least Squares Iterative Improvement Solution of Systems of Linear Equations with Equality Constraints - L2SLV ..399 Iterative Least Squares Solution of Banded Linear Equations - BLSQ ...................................403 Iterative Least Squares Solution of Sparse Linear Equations - SPLSQ, STLSQ ...........................405 Optimization Minimization of Functions of a Single Variable - FMIN ....407 Minimization of Functions of n Variable - OPTF ...........409 Unconstrained Minimum of the Sum of Squares of Nonlinear Functions Q LMDIFF ..................................411 Linear Programming - SMPLX, SSPLX ........................413 The Assignment Problem - ASSGN ...........................417 0-1 Knapsack Problem MKP .................................419 Transforms Inversion of the Laplace Transform - LAINV ...............421 Fast Fourier Transform - FFT, FFTl .......................425 Multivariate Fast Fourier Transform - MFFT, MFFTl ........427 Discrete Cosine and Sine Transforms - COSQI, COSQB, COSQF, SINQB, SINQF .................................429 Approximation of Functions Rational Minimax Approximation of Functions Q CHEBY ......433 Lp Approximation of Functions Q ADAPT ....................435 Calculation of the Taylor Series of Complex Analytic Function - CPSC, DCPSC .....................439 Curve Fitting Linear Interpolation - TRP ...............................443 Lagrange Interpolation Q LTRP ............................445 Hermite Interpolation - HTRP .............................447 Conversion of Real Polynomials from Newton to Taylor Series Form - PCOEFF ................................449 Least Squares Polynomial Fit - PFIT ......................451 Weighted Least Squares Polynomial Fit - WPFIT.............453 Cubic Spline Interpolation - CBSPL, SPLIFT ...............455 Weighted Least Squares Cubic Spline Fit - SPFIT ..........457 |Least Squares Cubic Spline Fitting with Equality and | Inequality Constraints - CSPFIT .....................459 Cubic Spline Evaluation - SCOMP, SCOMP1, SCOMP2 ..........461 Cubic Spline Evaluation and Differentiation - SEVAL, SEVAL1, SEVAL2 ..............................463 Integrals of Cubic Spline - CSINT, CSINT1, CSINT2 ........465 |Periodic Cubic Spline Interpolation - PDSPL ..............467 |Least Squares Periodic Cubic Spline Fitting - PDFIT ......469 |Periodic Cubic Spline Evaluation and Differentiation - | PSCMP, PSEVL ........................................471 N-Dimensional Cubic Spline Closed Curve Fitting - CSLOOP, LOPCMP, LOPDF ..............................473 Spline under Tension Interpolation - CURV1 ...............475 Spline under Tension Evaluation - CURV2 ..................477 Differentiation and Integrals of Splines under Tension Q CURVD, CURVI .......................................479 Two Dimensional Spline under Tension Curve Fitting - KURV1, KURV2 .......................................481 Two Dimensional Spline under Tension Closed Curve fitting - KURVP1,KURVP2 ............................483 Three Dimensional Spline under Tension Curve Fitting - QURV1, QURV2 .......................................485 |B-Splines ................................................487 |Finding the Interval that Contains a Point - INTRVL ......489 |Evaluation and Differentiation of Piecewise Polynomial | from its B-Spline Representation - BVAL .............491 |Evaluation of the Indefinite Integral of a Piecewise | Polynomial from its B-spline representation - BVALI..493 Conversion of Piecewise Polynomials from B-Spline to Taylor Series Form - BSPP ..........................495 Evaluation of Piecewise Polynomials from their Taylor Series Representation - PPVAL .......................497 Piecewise Polynomial Interpolation - BSTRP ...............499 |Weighted Least Squares Piecewise Polynomial Fitting - | BSLSQ ...............................................501 |Least Squares Piecewise Polynomial Fitting with | Equality and Inequality Constraints - BFIT ..........503 Surface Fitting over Rectangular Grids |Bicubic Splines and Bisplines under Tension ..............505 |Weighted Least Squares Bicubic Spline Fitting - SPFIT2 ...507 |Evaluation and Differentiation of Bicubic Splines - | CSURF, CSURF1, CSRF, CSRF2 ..........................509 Bispline under Tension Surface Interpolation - SURF ......513 Bispline under Tension Evaluation - SURF2, NSURF2 ........515 |Bivariate B-Spline Piecewise Polynomial Interpolation - | BSTRP2 ..............................................517 |Bivariate B-Spline Piecewise Polynomial Least Squares | Fitting - BSLSQ2 ....................................519 |Evaluation and Differentiation of Bivariate Piecewise | Polynomials from their B-Spline Representation - | BVAL2................................................521 Surface Fitting over Arbitrarily Positioned Data Points |Surface Interpolation for Arbitrarily Positioned | Data Points - TRMESH, GRADG, GRADL, SFVAL, SFVAL2 ...523 Manifold Fitting Weighted Least Squares Fitting with Polynomials of n Variables - MFIT, DMFIT, MEVAL, DMEVAL .............527 Numerical Integration |Evaluation of Integrals over Finite Intervals - | QAGS, QXGS, QSUBA, DQAGS, DQXGS .....................531 Evaluation of Integrals over Infinite Intervals - QAGI, DQAGI .........................................539 Evaluation of Double Integrals over Triangles Q CUBTRI ...543 Integral Equations Solution of Fredholm Integral Equations of the Second Kind - IESLV .......................................545 Ordinary Differential Equations/Initial Value Problems |The Initial Value Solvers - Introductory Comments ........549 Adaptive Adams Solution of Nonstiff Differential Equations - ODE .....................................551 |Adaptive Block RKF Solution of Nonstiff Differential Equations - BRKF45 ..................................555 Adaptive RFK Solution of Nonstiff Differential Equations - RFK45 ..................................559 Adaptive RFK Solution of Nonstiff Differebtial Equations with Global Error Estimation - GERK .................563 Adaptive Solution of Stiff Differential Equations - SFODE, SFODE1 .......................................567 Fourth-Order Runge-Kutta - RK ............................571 Eighth-Order Runge-Kutta - RK8 ...........................573 Partial Differential Equations Separable Second-Order Elliptic Equations on Rectangular Domains - SEPDE ...................................575 Discrete Random Number Generation |Uniform Random Selection of Values from a Finite Set of | Integers - URGET ....................................579 Continuous Random Number Generation |Uniform Random Number Generator - URNG, DURNG.............581 |Generating Points Uniformly in a Square - URNG2, DURNG2 ..583 |Generating Points Uniformly in a Circle - RCIR, DRCIR ....585 |Normal Random Number Generator - RNOR, DRNOR, | NRNG, DNRNG .........................................587 |Multivariate Normal Random Vector Generator - | NRVG, DNRVG, NRVG1, DNRVG1 ..........................589 |Exponential Random Number Generator - RANEXP, DRNEXP .....593 |Gamma Random Number Generator and the Chi-Square | Distribution - RGAM, DRGAM ..........................595 |Beta Random Number Generator - RBETA, DRBETA .............597 |F-Distribution Random Number Generator - FRAN, DFRAN .....599 |Student t-Distribution Random Number Generator - | TRAN, DTRAN .........................................601 |First Order Markov Random Number Generator - RMK1,DRMK1 ..603