An inventory of basic software for computer aided control system design (CACSD)

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system design (CACSD)

Citation for published version (APA):

Geurts, A. J. (1985). An inventory of basic software for computer aided control system design (CACSD). (WGS : report; Vol. 8501). Stichting Meet- en Besturingstechnologie, Werkgroep Programmatuur.

Document status and date: Published: 01/01/1985

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An

Inventory of Basic Software for

Computer Aided Control System Design

(CACSD)

Benelux Working Group on Software WGS-Report 85-1

Eindhoven University of Technology

Department of Mathematics and Computing Science

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Contents page

Introduction 3

Library index

4

Sources (libraries. packages)

10

Inventory

14

Explanation of the table entries

14

2. Mathematical routines

15

3.

Transformation routines

31

4.

Analysis routines

40

5.

Synthesis routines

46

6.

Data analysis

51

7. Identification

54

8.

Filter theory

58

Alphabetic index

59

AUTLIB

60

BIMAS

62

BIMASC

64

BYERS

66

DSP

67

KONTOS

69

USPACK

70

LPS 71 RASP 12 SUCE

16

SYCOT

78

TIMSAC

82

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-3-Introduction

This inventory is a next step on the way to the basic software library SYCOT for computer aided control system design (CACSD) to be realized by the Working Group on Software (WGS). The report on implementation and documentation standards 1) may be considered the first step.

It contains the subroutines. included in a number of libraries and packages. that may be considered as basic routines for solving problems in control and system theory. Also included are routines from some private collections. which we think are worthwhile to be mentioned. We did not screen the routines with respect to their quality. which. conse-quently. is not warranted for the included routines.

Not included are (main) programs. specific subroutines (so called nuclei) that are only used in other. more general. subroutines and machine dependent routines. Also not included are routines that belong to the chapter UTILITY ROUTINES.

Libraries or packages that are only commercially available. are left aside. However. the inclusion of a routine in the inventory does not mean that the routine is freely avail-able. Some packages are free. Others are free. or available against a nominal fee. for educa-tional use only.

Of course. the inventory is not complete. but nevertheless it gives an overview of what is available on basic software at this moment and. on the other hand. it reveals where possible gaps are.

The classification used is based upon the SLICE Library Index 2) and is problem-oriented. As a by-product the inventory gives also an idea of the relevance of this classification.

As it has been stated before. this inventory will be a starting point for the realization of the basic software library we are aiming at. Therefore. we would very much appreciate any comment on the classification and the contents of the inventory. Particularly. we will encourage anybody who knows about relevant software not included. to inform us. Com-ments should be adressed to

Mr. R.Kool. secretary WGS

Eindhoven University of Technology

Department of Mathematics and Computing Science Postbox 513

5600 MB Eindhoven The Netherlands

Finally. we gratefully acknowledge the help of Mr. L.G.F.C.van Bree and Mr. H.Willemsen in the preparation of the manuscript.

Eindhoven. May 1986 Working Group on Software

1) Working Group on Software. Implementation and Documentation Standards for the Basic Subroutine Library SYCOT. Eindhoven University of Technology. December 1983.

2) M.J.Denham. C.J.Benson. Implementation and Documentation Standards for the Software Library in

Control Engineertng (SLICE). Kingston Polytechnic. Control Systems Research Group, Internal Report

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Library index

1. UTILITY ROUTINES CUT) 1) 1.1. Text handling

1.2. File handling

1.3. Graphical input/output

104. General input/output routines (error messages) 1.5. Other utility routines

2. MATHEMATICAL ROUTINES (MA) 2.0. Auxiliary routines

2.0.1. Mathematical scalar routines

2.0.2. Mathematical vector/matrix routines 2.0.3. Sorting routines

2.004. Statistical routines 2.1. Linear algebra

2.1.1. Basic linear algebra manipUlations 2.1.2. Linear equations

2.1.3. Eigenvalues and eigenvectors

2.104. Decompositions and transformations 2.1.5. Matrix functions

2.2. Polynomial and rational function manipulations 2.2.1. Scalar polynomials

2.2.2. Scalar rational functions 2.2.3. Polynomial matrices 2.3. Optimization

2.3.1. Basic optimization routines 2.3.2. Unconstrained linear least squares 2.3.3.

2.304.

2.3.5. 2.3.6.

Unconstrained nonlinear least squares Minimax problems

4.2.

Polynomial Matrix Analysis (PM)

4.2.1.

Canonical and quasi canonical forms

4.2.2.

Equivalence transformations

4.2.3.

Greatest common divisor

4.2.4.

Continuous/discrete time

4.2.5.

Interconnection of subsystems

4.2.6.

Controllability. observability

4.2.7.

Inverse systems

4.2.8.

Poles. zeros

4.2.9.

Model reduction

4.2.10.

Root loci

4.2.11.

Nyquist diagrams

4.2.12.

Bode diagrams

4.3.

Rational Matrix Analysis CRM)

4.3.1.

Equivalence transformations

4.3.2.

Structural indices

4.3.3.

4.3.4.

4.3.5.

Inverse systems

4.3.6.

Poles. zeros

4.3.7.

Model reduction

4.3.8.

Root loci

4.3.9.

Nyquist diagrams

4.3.10.

Bode diagrams

4.4.

Frequency Response Analysis CFR)

4.4.1.

Polar/rectangular coordinates

4.4.2.

Interpolation

4.4.3.

Inverse systems

4.4.4.

4.4.5.

4.5.

Time Response Analysis (TR)

4.5.1.

Scaling

4.5.2.

Interpolation

4.5.3.

Convolution. deconvolution

4.5.4.

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-7-4.6. Markov Parameter Analysis (MP) 4.6.1. Scaling 4.6.2. Interpolation 4.6.3. Convolution. deconvolution 4.6.4. Interconnection of subsystems 4.6.5. Controllability. observability 4.6.6. Change of basis 4.6.7. Model reduction 4.7. Stability 5. SYNTHESIS ROUTINES

5.1. State Space Synthesis (SS)

5.1.1. Eigenvalue! eigenvector assignment 5.1.2. Riccati equations

5.1.3. Lyapunov equations 5.1.4. Sylvester equations 5.1.5. Minimum variance control 5.1.6. Dead beat control

5.1.1. Observers

5.1.8. Spectral factorization 5.1.9. Realization methods

5.1.10. Optimal regulator problems 5.1.11. Hierarchical control

5.1.12. Decentralized control 5.1.13. Non-interacting control 5.1.14. Model matching

5.2. Polynomial Matrix Fraction Synthesis (PM) 5.2.1. Eigenvalue!eigenvector assignment 5.2.2. Minimum variance control

5.2.3. Non-interacting control 5.2.4. Model matching

5.2.5. Parameter optimization

5.3. Rational Matrix Models Synthesis (RM) 5.4. Frequency Response Models Synthesis (PR) 5.5. Time Response Models Synthesis (TR) 5.6. Markov Parameter Models Synthesis eMP)

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6. DATA ANALYSIS (DA) 6.1. Scaling. interpolation 6.2. Statistical properties 6.3. Trend removal 6.4. Covariances 6.5. Spectra

6.6. Discrete Fourier transforms 6.7. Z-transforms 6.8. Prediction 6.9. Windowing 6.10. Filter design 7 . IDENTIFICATION (ID) 7.1. Nonparametric methods 7.1.1. Frequency analysis 7.2. 1.3. 7.1.2. Transient analysis Parametric methods 1.2.0. Auxiliary routines 7.2.1. Covariance methods

7.2.2. Deconvolution. numerical normal equations 7.2.3. Bayes estimation

7.2.4. Maximum likelihood 7.2.5. Least squares methods

1.2.6. Instrumental variable methods 1.2.1. Model reference methods 1.2.8. Prediction error methods 1.2.9. Stochastic approximation 1.2.10. Order/structure determination General methods 1.3.1. 7.3.2. 7.3.3. 7.3.4.

Parameter and state estimation combined Use of deterministic signals

Evaluation of input signals Test of model structure 8. FILTER THEORY CFT)

8.1. Kalman filters 8.2. LPC filters

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-9-9. ADAPTIVE CONTROL CAC) 9.1. Self-tuning control

9.1.1. Minimum variance methods 9.1.2. Predictive control methods 9.1.3. Pole placement methods 9.2. Model reference adaptive control 9.3. Parameter estimation

9.3.1. Matrix inversion lemma 9.3.2. Square root algorithm 9.3.3. UDU transformation 10. NONLINEAR SYSTEMS CNL) 10.1. Volterra series 10.2. Bilinear systems 10.3. Describing functions 10.4. Stability tests

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Sources Oibrariesp packages)

In this section a short description is given of the sources from which the routines are taken. If possible. literature is given for more detailed information.

1. AUn..m

A subroutine library for the design. analysis and simulation of control systems (Eidgenossische Technische Hochschule. ZUrich. Switzerland.).

2. BIMAS

A package of portable Fortran subroutines for solving several basic mathematical problems in CASAD.

Lit.

A.Varga. V.Sima. BIMAS - general description. Report ICI. TR-03.82. Central

Insti-tute for Management and Informatics. Bucharest. 1982.

A.Varga, V.sima. BIMAS - A Basic Mathematical Package for Computer Aided

Sys-tems Analysis and Design. Proceedings of the 9th IF AC Wodd Congress. Budapest.

Pergamon Press. 1985.

3. BIMASC (BIMAS CONTROL)

A package of Fortran subroutines for the analysis. modelling. design and simulation of control systems.

Lit.

A.Varga. BIMASC, general description. Report ICI. TR-10.83. Central Institute for

Management and Informatics. Bucharest. June 1983.

A.Varga. A. Davidoviciu. BIMASC - A Package of Fortran Subprograms for Analysis,

Modelling, Design and Simulation of Control Systems. Preprints of the 3rd IFAC

Symp. on CAD in Control and Engineering Systems. Copenhagen. July 31 - Aug. 2. 1985. Pergamon Press. 1985.

4. BLAS

Basic linear algebra subprograms. Lit.

C.L.Lawson. R.J.Hanson. D.R.Kincaid. and F.T.Krogh, Basic Linear Algebra Subpro-grams for Fortran Usage. ACM Trans. on Math. Software 5 (1979), 308-323.

S. BYERS

A collection of routines for solving optimal control problems. Lit.

R.Byers. Hamiltonian and Symplectic Algorithms for the Algebraic Riccati Equation.

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-11-6. DSP

IEEE-DSP package for discrete and fast Fourier transform. power spectrum analysis and correlation. fast convolution. FIR and IIR filters design and synthesis. cepstral analysis. interpolation and decimation.

Lit.

Programs for Digital Signal Prooossing. DSP Committee of the IEEE on ASSP. IEEE

Press. 1979.

7. EISPACK

A package for solving matrix eigenvalue problems. Lit.

B.T.Smith. J.M.Boyle. J.J.Dongarra. B.S.Garbow. Y.Ikebe. V.C.Klema. and C.B.Moler,

Matrix Eigensystem Routines - EISPACK Guide, Lecture Notes in Computer Science,

Vo1.6, Second Edition, Springer Verlag, New York, Heidelberg. Berlin. 1976.

B.S.Garbow. J.M.Boyle. J.J.Dongarra. C.B.Moler. Matrix Eigensystem Routines -EISPACK Guide Extension. Lecture Notes in Computer Science. Vol. 51, Springer

Ver-lag, Berlin. Heidelberg. New York, 1977.

8. EBLAS

An Extension to the Set of Basic Linear Algebra Subprograms. targeted at matrix vec-tor operations.

Lit.

J.J.Dongarra. J.Du Croz. S.Hammarling. and R.J.Hanson. A Proposal for an Extended Set of Fortran Basic Linear Algebra Subprograms. Argonne National Laboratory.

Mathematics and Computer Science Division. Technical Memorandum NoAl. December 1984.

9. KONTOS

APL programs for polynomial matrix manipulations. Lit.

A.Kontos. APL Programs for Polynomial Matrix Manipulations. Technical Report no

7913. december 1979. Rice University. Houston. Texas.

10. UNPACK

A package for solving systems of simultaneous linear algebraic equations. Lit.

J.J.Dongarra. J.R.Bunch. C.B.Moler. and G.W.Stewart. LINPACK Users Guide. SIAM

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11. USPACK

A collection of subroutines for analysis and synthesis of linear multivariable systems described in the state space.

Lit.

P.Hr.Petkov. N.D.Christov. M.M.Constantinov. A Program Package for

Computer-Aided Design of Digital Computer Control Systems. Preprint of SOCOC082. 217-220.

Madrid. Spain (1982).

12. LPS

Subroutines for Linear Prediction of Speech. Lit.

J.Markel and A.Gray. Linear Prediction of Speech. Springer Verlag. New York. 1976.

13. MINPACK

A package for the numerical solution of systems of nonlinear equations and nonlinear least squares problems.

Lit.

J.J.More. B.S.Garbow. K.E.Hillstrom. User Guide for MINPACK-l. Argonne National Laboratory. Report. ANL-80-74.

14. ODEPACK

A package for the solution of stiff and nonstiff systems of ordinary differential equa-tions.

L.it.

A.Hindmarsh. ODEPACK: a systematized collection of ODE solvers. in Scientific Com-puting: Applications of Mathematics and Computing to the Physical Sciences. R.S.Stepleman. Ed.

(IMACS Transactions on Scientific Computation. 10th IMACS World Congress. Mont-real 1982) North Holland Publ. Compo Amsterdam. New York. Oxford. 1983.

15. RASP

A library of Regulator Analysis and Synthesis Programs. Lit.

G.GrubeL Die regelungstechnische Programmbibliothek RASP. Regelungs-technik 31 (1983). 75-81.

16. SUCE

A Software Library In Control Engineering. Lit.

M.J.Denham. C.J.Benson. Implementation and Documentation Standards for the

Software Library in Control Engineering (SLICE). SEECS. Kingston Polytechnic.

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-

13-17. SSP

A set of computational subroutines for statistical or numerical problems in science and engineering.

Lit.

System/360 Scientific Subroutine Package, Programmer's Manual. Fourth edition. IBM

Technical Publication H20-0205-3. IBM Corporation. 1968.

18. SYCOT

A collective name for routines brought in by members of the WGS. It concerns mainly individual routines developed and used at the respective institutes of the members of the WGS. Information about these routines can be obtained via the secretary of the WGS.

19. TlMSAC

A program package for the analysis. prediction and control of time series. Lit.

H.Akaike. G.Kitagawa. E.Arahata. F.Tada. TIMSAC-78. Computer Science mono-graphs. No. 11, 1979. The Institute of Statistical Mathematics. Tokyo.

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Inventory

Explanation of the table entries.

The inventory of the collected subroutines is shaped in a table with six columns. with the following contents: Column 1: Column 2: Column 3: Column 4: Column 5: Column 6:

Section number. corresponding to the library index. or a subsection number. Short description of the problem that can be solved by the routine and/or the method used.

Type of input/output parameters or the type of arithmetic used. H different types are used. then the most significant type is mentioned. The following types are distinguished:

r real. single precision d real. double precision c complex. single precision z complex. double precision m mixed precision or types e extended precision

rid there is a real single and a real double precision routine with the same name.

integer

Name(s) of function(s) or subroutine(s). The effect of routines summed up together may be slightly different.

The source (library. package) where the routine is taken from.

A status indication of the routine(s) expressed by an integer value. The following indications are distinguished:

0: routine satisfies (almost) the SYCOT standards 1: standard Fortran code available

2: any implementation available 3: an algorithm available

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15

-2. MATHEMATICAL ROUTINES (MAl

2.0. Auxiliary routines

2.0.1. Mathematical scalar routines Decimal to binary conversion Relative unit roundoff quantity

Rounding a number to absolute or relative pre-cision

Conversion from integer to double precision Conversion of an array from single to double precision. vice versa

Square of the modulus of a complex number Modulus of a complex number

Complex division

Complex division in real arithmetic Complex square root

Polar from Cartesian coordinates

Angle in polar from Cartesian coordinates Divisibility test of two real numbers Entier of a real number

Mantissa and exponent of a real number

Errorfunction

2.0.2. Mathematical vector/matrix routines 2.0.3. Sorting routines

Sorts a vector in increasing order

Sorts a one-dimensional array

Rearranges a vector with a given permutation 2.0.4. Statistical routines

Normal (0. 1) random number generator Uniform random number generator

Uniform random number generator

t d rid d r d r r rid r d c d c r r d r r r d r r r d d c i d d d d d d r name source i BINARY TIMSAC 1 EPSLON EISPACK 1 DEPSLN BYERS 1 RUND RASP 1 DFLOAT BIMASC 1 ARRAY SSP 1 AMODSQ DSP 1 PYTAG EISPACK 1 SPYTAG SYCOT 1 DPYTAG SYCOT 0 COIV EISPACK 1 DCDIV BIMAS 1 CSROOT EISPACK 1 POLAR AUTLIB 1 ATAXY RASP 1 ATAXYD RASP 1 GDNS RASP 1 IKL RASP 1 ZPFORM RASP 1 ZPFORD RASP 1 ERF BYERS 1 SORTAG AUTLIB 1 SORTG DSP 1 SRTMIN TIMSAC 1 ORDNE2 RASP 1 COMPOR RASP 1 SORT RASP 1 PERMUT BIMAS 1 GRAND BYERS 1 RNOR TIMSAC 1 URAND BYERS 1 URAN RASP 1 RN TIMSAC 1 UM DSP 1

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Uniform random number generator both in r real and in bits

Uniformly distributed random numbers r t

Gaussian distributed random numbers d Generates an independent pair of random nor- r mal deviates

Generation of a pseudo random binary noise d sequence

Generation of a noise sequence with given mean d and variance name source RANBYT. Rl UNIF

DSP

RAND AUTLID GAUSS

RASP

NORMAL

DSP

PRBS

SYCOT

NOISE

SYCOT

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-

17-2.1. Linear algebra

2.1.1. Basic linear algebra manipulations 01. Auxiliary routines

Compatibility test of two matrices

Matrix storage from one- to two-dimensional array. vice versa

Linear dependency test of vectors Sum of the elements of an array

11. Elementary vector arithmetic

a. Scalar times vector plus vector

Scalar times vector

b. Innerproduct of two vectors

c. Maximum element of a vector Minimum element of a vector

Maximum-minimum element of a vector L1-norm of a vector

Mean of a vector L2-norm of a vector

Index of the largest component of a vector

Index of the largest vector component starting from a given index

d. Makes a copy of a vector

Swaps vectors t i r d d d r d c r d c r d m e c d d d d d r d c d r d c r d c d r d c r d r d c r

name

source

i FEHDIM RASP 1 ARRAYS RASP 1 ARRAY RASP 1 DPND RASP 1 MA11SM SYCOT 0 SAXPY BLAS 1 DAXPY BLAS 1 CAXPY BLAS 1 SSCAL BLAS 1 DSCAL BLAS 1

CSCAL. CSSCAL BLAS 1

SDOT BLAS 1

DDOT BLAS 1

DQooTA. DQooTI BLAS 1

DSooT. SDSooT BLAS 1

CooTC. CDOTU BLAS 1

MAXIND RASP 1 DMAX SYCOT 0 DMIN SYCOT 0 YMIN TIMSAC 1 MINMAX SYCOT 0 SASUM BLAS 1 DASUM BLAS 1 SCASUM BLAS 1 MEAN SYCOT 1 SNRM2 BLAS 1 SSSQ SYCOT 1 DNRM2 SYCOT 0 DSSQ SYCOT 0 SCNRM2 BLAS 1 ISAMAX BLAS 1 IDAMAX BLAS 1 ICAMAX BLAS 1 ORDROW SYCOT 0 SCOPY BLAS 1 DCOPY BLAS 1 CCOPY BLAS 1 XFR. SET DSP 1 COpy _TIMSAC ₁ SSWAP BLAS 1 DSWAP BLAS 1 CSWAP BLAS 1 EXCH DSP 1

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e. Sets each element of a vector to a constant value

Sets each element of a vector to zero

21. Elementary matrix-arltiunetic, square matrices

a. Sum. difference of matrices Scalar times matrix

b. Product of matrices

Product of matrices AB. ABT. ATB. ATbT Product AB. ABT. ATB. ATbT from A Band submatrices

Product of two real Schur matrices

Product of UT AU. A symmetric and U upper triangular

Product of XQXT. X arbitrary. Q symmetric

Product of XT QX. X arbitrary. Q symmetric

Product of ATBA. A and B arbitrary

Products ~AD or ~DA with ~ a real scalar. A arbitrary and D a matrix with ones down the minor diagonal

Transpose of a matrix c. L1-norm of a square matrix

Frobenius norm of a square matrix Ll-norm of a symmetric matrix Frobenius norm of a symmetric matrix Measure of the difference of two matrices Maximum element of a matrix

d. Makes a copy of a matrix Composition of blockmatrices

Composition of matrices. column-wise Composition of matrices. row-wise Composition of matrices

Composition of matrices. double to single pre-cision

Partitioning of a matrix Modification of a (sub)matrix

e. Initialization of a matrix by a unity matrix Sets diagonal elements of a matrix

22. Elementary matrix-aritlunetic, red:angular

matrices

a. Sum or difference of arbitrary matrices Sum of matrices Difference of matrices t d r d r r d d d d r r r r d d d d d r d d d d r d rId d d d rId d d d r r d d r r d r name source EQROW SYCOT I ZERO DSP

-MSCALE RASP SQAXB SYCOT I MATM SLICE

MULT. MAMUDD RASP

AMTM RASP EMULSH BIMAS UTAU BIMAS MXQXT AUTLm MXTQX AUTLffi ATSA SYCOT ATBA SYCOT DAD BIMAS TRANP RASP TRPS SYCOT FNRMl BYERS FROB BYERS NORM AUTLIB SNRMl BYERS SYFROB BYERS ITERR RASP MAXEL RASP NORMM SYCOT EQUATE RASP

INSEDS. INSERT RASP

JUXTC RASP JUXTR RASP MASEDD RASP MASEDS RASP PART RASP MAKODD RASP UMTY RASP HHUNIT AUTLffi DCLA SSP APMB SYCOT ADD RASP SMADD AUTLIB MADD.GMADD SSP SUBT RASP MSUB.GMSUB SSP

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-

19-Scalar times matrix

Matrix plus constant times unity matrix b. Product of matrices

Product of a symmetric and arbitrary matrix Product: AT B. A and B arbitrary

Product: AB with A and B such that AB is

symmetric

Product:

AIl'.

A and B arbitrary Product: matrix and its transpose Product: matrix and triangular matrix Interchanges two rows of a matrix Transpose of a matrix

Product: ASAT. S symmetric

c. Norms of matrices. Lp. p-l. 2. co. Frobenius Frobenius norm of the difference of two matrices

Frobenius norm of an arbitrary matrix d. Copies a matrix

Copies a part of a matrix

Copies a column of a matrix into a vector Copies a row of a matrix into a vector Vertical partioning of a matrix Horizontal partioning of a matrix

Exchanges two rows/columns of a matrix e. Initialization of null-matrix

Sets each element of a matrix to a given scalar Annulates a part of a matrix

f. Trace of a matrix

31. EletMntary matrix-vector arithmetic

a. Matrix times vector plus vector

aAx + y. a scalar. A general matrix. x and y vectors

Idem. A general band matrix

Idem. A symmetric matrix . Idem. A Hermitian matrix

Idem. A symmetric bandmatrix

t r d d d r d d r r r r d r r r d d r d d d r r r r r r r d r r d d d r d c z r d c

z

r d c z r d name source i SMPY SSP 1 DIADD RASP 1

AMI'M. MAMUDD RASP 1

MULT RASP 1 MPRD.GMPRD SSP

1

AXB SYCOT 0 MULTSF BYERS 1 TPRD.GTPRD SSP 1 ABCS AUTLIB 1 GTAPB SSP 1 MATA SSP 1 ATA SYCOT 0 MTDS SSP 1 RINT SSP 1 GMTRA SSP 1 TRANSP SYCOT 0 PAPT SYCOT 0 NORMSS RASP 1 NORMS1 RASP 1 DCNORM SYCOT 1 FNORM SYCOT 0 MCPY SSP 1 XCPY SSP 1 CCPY SSP 1 RCPY SSP 1 RCUT SSP 1 CCUT SSP 1 CHANGE SSP 1 MNUL2D RASP 1 SCLA SSP 1 NULL AUTLIB 1 TRCE RASP 1 TRACE SYCOT 0 MULVA BIMASC 1 SGEMV EBLAS 4 DGEMV EBLAS 4 CGEMV EBLAS 4 ZGEMV EBLAS 4 SGBMV EBLAS 4 DGBMV EBLAS 4 CGBMV EBLAS 4 ZGBMV EBLAS 4 SSYMV.SSPMV EBLAS 4 DSYMV. DSPMV EBLAS 4 CHEMV. CHPMV EBLAS 4 ZHEMV. ZHPMV EBLAS 4 SSBMV EBLAS 4 DSBMV EBLAS 4

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t name source i Idem. A Hermitian band matrix

I

~

CHBMV ZHBMV BBLAS BBLAS 4 4

b. Triangular matrix times vector STRMV. STPMV BBLAS 4

d DTRMV. DTPMV BBLAS 4

c CfRMV. CfPMV BBLAS 4

z ZTRMV. ZTPMV BBLAS 4

Triangular band matrix times vector r STBMV BBLAS 4

d DTBMV BBLAS 4

c CTBMV BBLAS 4

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-

21-2.1.2. Linear equations

01. Auxiliary routines

Test on definiteness of a matrix

Linear independency of rows/columns of a matrix

Rank of a matrix

11. Solution,full matrix

a. Arbitrary matrix, decomp. given

. Arbitrary matrix. decomp. not given b. Positive definite matrix. decomp. given

c. Symmetric matrix. decomp. given

d. Hermitian matrix. decomp. given e. Triangular matrix. decomp. given

12. Solution bandmatrix

a. Arbitrary matrix. decomp. given

Idem. decomp. not given

b. Positive definite matrix. decomp. given

c. Symmetric matrix. decomp. not given

f. Tridiagonal matrix. decomp. not given

Tridiag. pos def. matrix. decomp. not given

I

t d r d r d c z r r r d c z d r d c z c z r d c z r d c z rid r d c z rid r d c z r d c z name source i DEFIT RASP 1 MFGR SSP 1 RANK SYCOT 1 SGESL LINPACK 1 DGESL LINPACK 1 CGESL LINPACK 1 ZGESL LINPACK 1 SIMQ SSP 1 CROUT SYCOT 1 SPOSL.SPPSL LINPACK 1 DPOSL. DPPSL LINPACK 1 CPOSL. DPPSL UNPACK 1 ZPOSL.ZPPSL LINPACK 1 FACTOR RASP 1 SSISL. SSPSL LINPACK 1 DSISL. DSPSL LINPACK 1 CSISL. SSPSL LINPACK 1 ZSISL. ZSPSL LINPACK 1 CHISL. CHPSL LINPACK 1 ZHISL. ZHPSL LINPACK 1 STRSL LINPACK 1 DTRSL LINPACK 1 CTRSL LINPACK 1 ZTRSL LINPACK 1 SGBSL LINPACK 1 DGBSL LINPACK 1 CGBSL LINPACK 1 ZGBSL LINPACK 1 BANDV EISPACK 1 SPBSL LINPACK 1 DPBSL LINPACK 1 CPBSL LINPACK 1 ZPBSL LINPACK 1 BANDV EISPACK 1 SGTSL LINPACK 1 DGTSL LINPACK 1 CGTSL LINPACK 1 CGTSL LINPACK 1 SPTSL LINPACK 1 DPTSL LINPACK 1 CPTSL LINPACK 1 ZPTSL LINPACK 1

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13. Solution Hessenberg matrix

Solution. matrix with two nontrivial lower subdiagonals

Solution. matrix with three nontrivial lower subdiagonals

14. Solution triangular matrix

a. Arbitrary upper or lower triangular matrix

b. Idem. triangular band matrix

31. Inverse and determinant of a full matrix

a. Arbitrary matrix

Inverse only

b. Positive definite matrix

Inverse only c. Symmetric matrix

d. Hermitian matrix e. Triangular matrix

Inverse only

32. Inverse and determinant of a bandmatrix

a. Arbitrary matrix

b. Positive definite matrix

35. Generalized inverse of a general matrix

t d d d r d c z r d c z r d c z d r d d r d c z d r d c z c z r d c z d r d c z r d c z d name source i HSLV BIMAS 1 H2SLV BIMAS 1 H3SLV BIMAS 1 STRIV.STPIV EBLAS 4 DTRIV.DTPIV EBLAS 4 CTRIV,CTPIV EBLAS 4 ZTRIV,ZTPIV EBLAS 4 STBIV EBLAS 4 DTBIV EBLAS 4 crBIV EBLAS 4 ZTBIV EBLAS 4 SGEDI UNPACK 1 DGEDI UNPACK 1 CGEDI UNPACK 1 ZGEDI UNPACK 1 INV RASP 1 MINV SSP 1 INVDET TIMSAC 1 INVMAT SYCOT 1

SPODI. SPPDI UNPACK 1

DPODI. DPPDI UNPACK 1

CPODI.CPPDI UNPACK 1

ZPODI. ZPPDI UNPACK 1

SMINVD DSP 1

SSID!. SSPDI UNPACK 1

DSIDI. DSPDI UNPACK 1

CSIDI. CSPDI LINPACK 1

ZSIDI. ZSPDI UNPACK 1

CHID!. CHPDI UNPACK 1

ZHIDI, ZHPDI UNPACK 1

STRIDI UNPACK 1

DTRIDI UNPACK 1

crRIDI UNPACK 1

ZTRIDI LINPACK 1

INVERS. TRIINV TIMSAC 1

SGBDI UNPACK 1 DGBDI UNPACK 1 CGBDI UNPACK 1 ZGBDI UNPACK 1 SPBDI UNPACK 1 DPBDI UNPACK 1 CPBDI UNPACK 1 ZPBDI LINPACK 1 INVERS SYCOT 1

(24)

-

23-t name source i

51. Matrix equations

Solution of a homogeneous or inhomogeneous rid DMFGR SSP 1

matrix equation with arbitrary matrix

Solution of a homogeneous equation d LOESHO RASP 1 Solution of an inhomogeneous equation d LOESIN RASP

1

d LUSLV BIMAS 1

Solution of a matrix equation with an arbi- r GELG SSP 1 trary matrix

Solution of a matrix equation with a positive d SYMPDS RASP 1 definite matrix

Solution of a matrix equation with a sym- r GELS SSP 1 metric matrix

Solution of an inhomogeneous equation with a d SOLHES BIMAS 1 Hessenberg matrix, decomposed by DECHES

Solution of an inhomogeneous matrix equation d SOLVE TIMSAC 1 with an upper triangular matrix

91. Condition of a triangular matrix r STRCO UNPACK. 1

d DTRCO UNPACK. 1

c CTRCO UNPACK 1

(25)

t name 2.1.3. Eigenvalues and eigenvectors

02. Specific transformations

a. Balances an arbitrary matrix and isolates eigen- rid BALANC values. whenever possible

clz Balances an arbitrary matrix in order to minim- r

ize its maximum norm

Isolates eigenvalues (mod. of BALANC. d EISPACK)

Decodes and applies the transformation of d BALANC

Orders the eigenvalues of a quasi-triangular d matrix CBAL BALRS PERMUT UNBAL QLORDR d SEORL SEOR2 Orders the eigenvalues of a real Schur matrix r SORT

b. One implicit QR step on an upper Hessenberg matrix d QRSTEP A single QR step A single QL step r QRSTEP d QLSTEP c. Arbitrary (sub)matrix to upper Hessenberg

form

rid ELMHES.ORTHES

d.

r

c/z

Arbitrary matrix to lower Hessenberg form d r Arbitrary (sub)matrix or upper Hessenberg r form to quasi-triangular form

Hessenberg form to Schur form with ordered d eigenvalues

Hessenberg form to Schur form with transfor- d mation matrix

Real Schur decomposition of a real upper d Hessenberg matrix (mod. of HQR2. EISPACK)

d

Lower Hessenberg matrix to lower quasi- d triangular matrix

Lower triangular Schur decomposition d Real Schur decomposition (mod. of RG. d EISPACK)

Schur decomposition of a 2 X 2 matrix d Reduction of a 2 X 2 diagonal block of a real r Schur matrix to upper triangular form

Splits a 2 X 2 diagonal block of an upper d quasi-triangular matrix

Real Schur form to block-diagonal form d Hermitian matrix to (real) symmetric tridiago- rid nal matrix HSHLDR COMHES. CORTH LOWHES HESSCO QTRORT HQR3 HQRT HQRIT HQRl. HQR4 QLIT LSCHUR RSCHUR SCHUR2 SPLIT SPLIT BDIAG TREDL TRED2 TRED3

Arbitrary tridiagonal matrix to symmetric tri-diagonal matrix

c/z HTRIDI, HTRID3 rid FIGI. FlGI2

source EISPACK EISPACK SLICE BYERS BYERS BYERS BIMAS SYCOT BIMAS SYCOT BYERS EISPACK SYCOT EISPACK BYERS AUTLIB SLICE RASP SYCOT BYERS BIMAS BYERS BYERS BYERS BYERS SYCOT BIMAS BIMAS EISPACK EISPACK EISPACK EISPACK

(26)

(27)

14. Eigenvalues andlor eigenvectors of an upper Hessenberg matrix

t

All eigenvalues rid

clz

All eigenvalues and eigenvectors rid

r

c/z

All eigenvalues and eigenvectors (mod. of d HQR2. EISPACK)

Some eigenvectors rid

15. Eigenvalues andlor eigenvectors of a Schur

matrix

c/z

All eigenvalues d

All eigenvectors d

16. Accuracy test of eigenvalues and eigenvectors

21. Transformation of eigenvectors from reduced

problem to original problem

original matrix reduced matrix arbitrary balanced

arbitrary upper Hessenberg

Hermitian symmetric tridiagonal tridiagonal symmetric tridiagonal

d rid clz rid

clz

r rid clz rid Backtransformation of Schur vectors from per- d muted (PERMUT -BYERS. see 02a) to original matrix

22. Transformation of eigenvectors of a general

eigenvalue-problem name HQR COMLR. COMQR HQR2 HQRT COMLR2. COMQR2 HQR2 INVIT CINVIT SEIG SVEC EITEST BALBAK CBABK2 ELMBAK. ORTBAK COMBAK. CORTB BCKMLT TRBAK1. TRBAK3 HTRmK. HTRm3 BAKVEC PRMBAK

Symmetric general eigenvalue-problem reduced rid REBAKB. REBAK to a symmetric standard eigenvalue-problem

31. General eigenvalue-problem, full matrices

a. Arbitrary matrices. all eigenvalues and eigen- rid ROO vectors Cif desired)

b. Symmetric and positive definite matrices. all rid RSG eigenvalues and eigenvectors (if desired)

c. Variants of the general eigenvalue-problem

ABx =< AX and BAx

=

AX. A symmetric. B posi- rid RSGAB. RSGBA

tive definite, all eigenvalues and eigenvectors Cif desired) source EISPACK EISPACK EISPACK SYCOT EISPACK RASP EISPACK EISPACK BIMAS BIMAS RASP EISPACK EISPACK EISPACK EISPACK SYCOT EISPACK EISPACK EISPACK BYERS EISPACK EISPACK EISPACK EISPACK

(28)

-

27-t name source i

32. Reduced general eigenvalue-problem

Quasi-triangular and triangular matrix, some rId QZVEC eigenvectors

EISPACK 1

d

33. Generalized eigenvalue-problem, singular pencils Computes Kronecker indices and all elementary r divisors of an M by N pencil AB - A

Computes the Kronecker row indices and the r infinite elementary divisors of an M by N pencil AB-A

Computes the Kronecker column indices and the r infinite elementary divisors of an M by N pencil AB-A

41. Invariant subs paces

Reordering of Schur form for invariant subspace r with prescribed spectrum

42. Deflating subspaces

r

d d

Reordering of generalized Schur form for r deflating subspace with prescribed spectrum

51. Hamiltonian systems

d d

Reduction to Hamiltonian - Hessenberg form d

Hamiltonian QR iteration d

Hamiltonian QR step d

Hamiltonian-&hur decomposition d Hamiltonian matrix to square reduced Hamil- d tonian matrix

Orders the eigenvalues of a Hamiltonian tri- d angular matrix

61. Conditioning, estimates

Computes the condition number of an eigen- r value Estimates sep(TT. -T) d QZVECM SSXKF MRINX MCINX ORDERS EXCHNG. SWAPP EXCHQR. QRSTEP INVSUB. EXCHQR EXCHNG ORDERZ EXCHQZ, QZSTEP DDSUBS. DEXCHQ EXCQZS HAMHES HAMIT HAMQR HAMSCH SQRED ORDER BIMAS 1 SLICE 1 SLICE 1 SLICE 1 LISPACK 1 SYCOT 1 LISPACK 1 SYCOT 0

BIMAS

1

LISPACK 1 LISPACK 1 SYCOT 0 BIMAS 1 BYERS BYERS BYERS BYERS BYERS BYERS 1 1 1 1 1 1

CONDIT. HQRNOZ LISPACK 1

(29)

t name 2.1.4. Decompositions and transformations

01. Auxiliary routines

Update of a QR or Cholesky decomposition r SCHUD, SCHDD SCHEX d DCHUD, DCHDD DCHEX c CCHUD, CCHDD CCHEX z ZCHUD, ZCHDD ZCHEX

rid R1UPDT, RWUPDT Update of a Cholesky decomposition d LDLT

02. Elem£ntary transformations

a. Constructs a Householder transformation r SNREFG d DNREFG Applies a Householder transformation r SNREF

d DNREF

Constructs and applies a Householder transfor-mation

d H12

d

Householder reduction d

d

Constructs a reflection of length 2 or 3 d

Constructs a reflection d

Symmetric similarity transformation by a d reflection of length 2 or 3

Symmetric similarity transformation by a d reflection

Applies a reflection of length 2 or 3 to a set of d vectors

Applies a reflection to a set of vectors d Constructs a skew Householder reflection d Applies a skew Householder reflection d b. Constructs a Givens plane rotation r

d

Applies a Givens plane rotation r d

d

c. Applies the transformation of ORTHES d (EISPACK)

Applies the transformation of ELMHES d (EISPACK) H12 MREDCT REDUCT G3REF GENREF S3REF SYMREF V3REF VECREF DSREFG DSREF SROTG GIV SROTMG DROTG DGIV DROTMG G1 SROT SROTM DROT DROTM G2 HSHMLT ELTR source UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK MINPACK RASP SYCOT SYCOT SYCOT SYCOT RASP BIMASC TIMSAC TIMSAC BYERS BYERS BYERS BYERS BYERS BYERS SYCOT SYCOT BLAS SYCOT BLAS BLAS SYCOT BLAS RASP BLAS BLAS BLAS BLAS RASP BIMAS BIMAS

(30)

-

29-03. Accumulation of elementary transf0171Ultions

Similarity transformation Orthogonal transformation

04. Rank-J update

a. A + OlXyr • A general matrix

A

+

OlXyli. A general matrix b. A + OlXXT • A symmetric matrix c. A

+

Oln!l. A Hermitian matrix

05. Rank-2 update

a. A + OlXyr + Olyr. A symmetric matrix b. A + OlXyR + Ciyx!l. A Hermitian matrix

11. Matrix decomposition, full matrix

a. Arbitrary matrix. LR decomposition

b. Positive definite matrix Cholesky decomp.

c. Symmetric matrix. UDuH decomposition

Idem. QTQT decomposition

Idem. Cholesky decomposition. semi definite

I

t rid d rid d rid r d c z c z r d c z r d c z r d c z r d c z d d d r d c z rid d r d name source i ELTRAN EISPACK 1 ELTRN BIMAS 1 ORTRAN EISPACK 1 ORTR B1MAS 1 QFORM MINPACK 1 SGERl EBLAS 4 DGERl EBLAS 4 CGER1U EBLAS 4 ZGER1U EBLAS 4 CGER1C EBLAS 4 ZGERIC EBLAS 4 SSYR1.SSPR1 EBLAS 4 DSYR1.DSPRl EBLAS 4 CHER1.CHPR1 EBLAS 4 ZHER1.ZHPR1 EBLAS 4 SSYR2.SSPR2 EBLAS 4 DSYR2.DSPR2 EBLAS 4 CHER2.CHPR2 EBLAS 4 ZHER2.ZHPR2 EBLAS 4

SGECO. SGEFA UNPACK 1

DGECO.DGEFA UNPACK 1

CGECO.CGEFA UNPACK 1

ZGECO.ZGEFA UNPACK 1

SPOCO. SPOFA UNPACK 1

SPPCO. SPPFA UNPACK 1

SCHDC UNPACK 1

DPOCO.DPOFA UNPACK 1

DPPCO.DPPFA LINPACK 1

DCHDC UNPACK 1

CPOCO.CPOFA UNPACK 1

CPPCO. CPPFA UNPACK 1

CCHDC UNPACK 1

ZPOCO.ZPOFA UNPACK 1

ZPPCO. ZPPFA UNPACK 1

ZCHDC UNPACK 1

FACTOR RASP 1

SYMPDS RASP 1

LTINV TIMSAC 1

SSICO. SSIFA UNPACK 1

DSICO. DSIFA UNPACK 1

CSICO. CSIFA UNPACK 1

ZSICO. ZSIFA UNPACK 1

TRED2 EISPACK 1

DFASI SYCOT 1

CHOLD SUCE 1

(31)

d. Hermitian matrix. UDuH decomposition

12. Matrix decomposition, bandmatrix

a. Arbitrary matrix. LR decomposition

b. Positive definite matrix. Cholesky decomp.

c. Symmetric matrix. QTej" decomposition

13. Matrix decomposition, Hessenberg matrix

LR decomposition

21. QR factorization of a rectangular matrix

Transformation of a matrix into a triangular matrix

Performs the Householder transformation QR factorization with column permutation RQ decomposition of a square matrix Modified Gram-Schmidt algorithm

31. Singular value decomposition

name c CHICO. CHIFA CHPCO.CHPFA z ZHICO. ZHIFA ZHPCO.ZHPFA r SOBCO. SGBF A d DOBCO. DGBFA c CGBCO.CGBFA

z

ZGBCO.ZGBFA r SPBCO. SPBF A d DPBCO.DPBFA c CPBCO.CPBFA z ZPBCO.ZPBFA r MFSD rid BANDR d DECHES r SQRDC. SQRSL d DQRDC. DQRSL c CQRDC.CQRSL z ZQRDC.ZQRSL ? QRFAC r SMORTH r HOTRAN d d r d d HUSHLD HUSHLI HOUTRA SQRQDC MGSA

Arbitrary rectangular matrix r SSVDC

DSVDC CSVDC ZSVDC MINFlT. SVD SNVDEC MSVD d c z rid d d

Large matrix with low rank d

41. Transformation by multiplication

Pre- or postmultiplication of an arbitrary r matrix by an orthogonal matrix

Multiplication of a matrix by a product of rid Givens rotations 42. Equivalence transformation Symmetric matrix d ASVD HHDME. HHDML RIPMYQ SYMEQU source UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK UNPACK SSP EISPACK BIMAS LINPACK UNPACK I LINPACK UNPACK MINPACK AUTLID AUTLIB TIMSAC TIMSAC AUTLIB BIMASC TIMSAC LINPACK UNPACK LINPACK LINPACK EISPACK RASP TIMSAC SYCOT SLICE MINPACK BYERS

(32)

-

31-t

2.1.5. Matrix functions

11. Matrix exponen:tial

Matrix exponential of a real non-defective r matrix

Matrix exponential and its integral r

d

r Matrix exponential and integrals of it d

21. Exponential of a matrix

Computes the exponential of a matrix by block d diagonalization and rational Pade approxima-tions

Computes the exponential of a matrix by d rational Pade approximations

Exponential of an arbitrary matrix r Computes the matrix exponential with accu- r racy estimate

Computes the exponential of a real Schur d matrix by rational Pade approximations

name MEEIG MEINT EAT2 LSP2 EAT4 BPADE PADE MEPAD PADE PADES source SLICE SLICE RASP AUTLIB RASP BlMAS BIMAS SLICE LISPACK BIMAS i 1 1 1 1 1 1 1 1 1 1

(33)

(34)

-

33-t name source i

2.3. Optimization

2.3.1. Basic optimization routines

01. Consistency check of a Jacobian matrix rid CHKDER MINPACK 1

11. Evaluates the gradient of a function r GUNC AUTLIB 1

Evaluates the gradient of a function and res- r GCON AUTLIB 1 trictions

Computes an approximation to the gradient by d SGRAD TIMSAC 1 differentiation

Forward difference approximation of a square rid FDJACl MINPACK 1 Jacobian matrix

Forward difference approximation of a rec- rid FDJAC2 MINPACK 1 tangular Jacobian matrix

21. Line minimization

Onedimensional minimum along a given direc- d MINF RASP 1 tion

Local minimum along a given direction r UNIOP AUTLIB 1

Direction for line minimization rid LMPAR. QRSOLV MINPACK 1

rid DOGLEG MINPACK 1

Line search via parabolic interpolation r LINE SYCOT 1

Linear search along a given direction d LINEAR TIMSAC 1 2.3.2. Unconstrained linear least squares

Least squares solution of an over- or under- d DQRSLT BIMASC 1

determined linear system

Solution for a full rank arbitrary matrix r LLSQ SSP 1

Solution. QR-factorization given r SQRSL LINPACK 1

d DQRSL LINPACK 1

c CQRSL LINPACK 1

z ZQRSL LINPACK 1

Minimal LLS-solution. arbitrary matrix r LINMIN AUTLIB 1

Total LLS-solution of AX=B, with A and B r STLLS SYCOT 0 inaccurate

d DTLLS SYCOT 0

Adaptive LS solution _d ASOLVE SYCOT 2

2.3.3. Unconstrained nonlinear least squares

Solution. Jacobian matrix required rid LMDER. LMDERl MINPACK 1

rid LMSTR, LMSTRl MINPACK 1 Solution. Jacobian matrix not required rid LMDIF. LMDIF1 MINPACK 1

2.3.4. Minimax problems

Discrete piecewise linear minimax approxima- d DPLMMA SYCOT 0 tion

(35)

2.3.5. Other unconstrained problems

11. Scalar functions

Minimum of a scalar function r

Minimum of a scalar function in a predeter- r mined interval

12. Multi-variable functions

Local minimum of a multivariable function r r Global minimum. gradient not required r t

d

r Global minimum. gradient required d

2.3.6. Linearly constrained linear least squares Solution of a 11s problem

Solution of a nonnegative lis problem

2.3.7. Linearly constrained nonlinear least squares

2.3.8. Other linearly constrained problems

11. Linear programming 21. Quadratic programming d d d Quadratic programming d

31. Non linear programming

Projected gradient method with upper and r lower limits

2.3.9. Nonlinearly constrained nonlinear least squares

2.3.10. Other nonlinearly constrained problems

Nonlinear mathematical programming problem r Sequential linear least squares programming to d solve a general nonlinear optimization problem

name OPT1. INTSUC GOLSEC FMCG1. FMFPl FMFP BOX POWELL UNCOP OFP FMIN LSQ NNLS LOP. LOPEI GPLIN

NLP

SLLSPQ source AUTLIB AUTLIB SSP SYCOT SYCOT RASP AUTLIB RASP RASP RASP RASP RASP SYCOT AUTLIB RASP

(36)

-

35-t name

source

i

24. Zeros and nonlinear equations 2.4.1. Zeros of a polynomial

Computes the zeros of a quadratic function d QUAD RASP 1 Computes the zeros of a real polynomial d ZRPOLY RASP 1

d RPOLY SYCOT 0

Computes the zeros of a real polynomial by d POLYRT TIMSAC 1 Newton-Raphson

2.4.2. Zero(s) of a function

2.4.3. Systems of nonlinear equations

Ol. Auxiliary routines

Consistency check of a Jacobian matrix rid CHKDER MINPACK 1

Approximation of a Jacobian matrix rid FDJACI MINPACK 1

Direction for line minimization rid DOGLEG MINPACK 1

11. Solution. Jacobian matrix not required rid HYBRD.HYBRDI MINPACK 1 Solution. Jacobian matrix required rid HYBRJ. HYBRJi MINPACK 1

(37)

t

2.5. Differential equations 2.5.1. Initial value problems

01. Auxiliary rautines

Symbolic LDU-factorization of a sparse matrix d Symbolic LDU-factorization of a sparse matrix d and solution of the system of linear equations Solution of a system of linear equations with d sparse matrix. LDU-factorization given

Solution of the transpose system. LDU- d factorization given

11. System of first order differential equations. r fixed step integration

System of first order differential equations r r

21. Explicit systems of first order equations

Solution of a stiff or nons tiff system. with d automatic switch between stiff and nons tiff methods

Idem. with additionally determination of roots of constraint functions

Solution for either a stiff or a nonstiff system Idem. but intended for problems with a sparse Jacobian matrix (when the problem is stiff) Solution of a nonstiff or mildly stiff system Solution of a stiff system of first order equa-tions

22. Linearly implicit systems of first order equations

d d d d d name NSFC NNFC NNSC NNTC RUNU

RK58

EULER RK LSODA LSODAR LSODE LSODES RKF45 INTGRA

Solution of either a stiff or a nonstiff system d LSODI 2.5.2. Boundary value problems

2.5.3. Partial differential equations

source i ODEPACK 1 ODEPACK 1 ODEPACK 1 ODEPACK 1 AUTLIB 1 AUTLIB 1 AUTLIB 1 AUTLIB 1 ODEPACK 1 ODEPACK ODEPACK ODEPACK BIMASC SYCOT 1 1 1 1 1 ODEPACK 1

(38)

-

37-3. TRANSFORMATION R.OUTINES

t nam.e source i 3.1. State space

11. Computes the complex frequency response matrix c SSXFR SLICE

1

Reduces a time-invariant multi-input system to r SSXMC SLICE 1 orthogonal canonical form

Reduces a time-invariant single-input system to r SSXSC SLICE 1 orthogonal canonical form

Computes controller Hessenberg form d DCHESS SYCOT 0 12. Finds the transfer function matrix of a given ssr r SSXTM SLICE 1 Finds the transfer function matrix of a given ssr, d TSMT BIMASC 1 using orthogonal transformations

Finds the transfer function matrix of a given ssr. d TSMTI BIMASC 1 using stabilized elementary similarity

transfor-mations

13. Finds a minimal ssr (staircase form) for a given r SSXMR SLICE 1 ssr

Finds a relatively prime left or right pmr which r SSXPM SLICE 1

is equivalent to a given ssr

21. Finds the dual (transpose) system linear time-invariant ss-model

of a given r SSXDL SLICE 1

d DUALS BIMASC 1

31. Rational transfer function of a ssr d TRANSF SYCOT 1 Determines a reduced order ss-model from a d REMIN BIMASC 1 non-minimal ss-model

Transfer matrix from poles and residues rid TFEIG RASP 1 Inverse of sI - A and the transfer matrix of a d FADDEE SYCOT 1

given ss-model

Minimal realization of a transfer matrix by the r NRELDI AUTLm 1 method of Nour-Eldin

41. Rational transfer function to system matrix form d ZUSTD RASP 1 Determines a non-minimal. uncontrollable and d RENEM BIMASC 1 unobservable ssr for a given transfer matrix

Determines a non-minimal. controllable ssr for a d RENEMC BIMASC 1 given transfer matrix

Determines a non-minimal. observable ssr for a d RENEMO BIMASC 1 given transfer matrix

Ssr in phase variable canonical form from a given r PHAVA AUTLm 1 transfer function (SIS0)

51. Computes the ssr for the cascaded interconnec- r SSCASC SLICE 1 tion of two ss-systems

Computes the ssr for the feedback interconnec- r SSFEED SLICE 1 tion of two ss-systems

Computes the ssr for the parallel interconnection r SSPARA SLICE 1 of two ss-systems

(39)

t

Connection of an internal model to the output of d an open-loop state space system

61. Staircase form d

Reduction to staircase form with triangular d pivots

r 3.2. Generalized state space

3.3. Polynomial matrix fractions

Finds the dual right(Ieft) polynomial matrix r representation (pmr)

d

Computes the transfer matrix of a left or right c pmr at a given frequency

Finds a ssr equivalent to a given left or right pmr r 3.4. Polynomial matrix quadruples

3.5. Rational transfer functions

Computes the value of a complex valued rational r transfer function for a given frequency (SISO) Finds a relatively prime left or right pmr for a r given proper transfer function matrix (MIMO) Finds a minimal ssr for a given proper transfer r function matrix

Sampled-data system corresponding to a continu- d ous system given by a transfer matrix

3.6. Frequency response name EXTI TSCO DSTAIR DECZR PMXDL TFFAHN PMXFR PMXSS TFXFR TMXPM TMXSS TMTCD

Real and imaginary part of a matrix frequency rid TFLAUB response

3.7. Time response (impulse, step response, etc.)

Output sequence of a given ssr d Output sequence of a given ssr with a Hessenberg d matrix 3.8. Markov parameters a. Auxiliary routines SSOUT SSOUT2 source i BIMASC 1 BIMASC 1 SYCOT 0 SLICE SLICE RASP SLICE SLICE 1 1 1 1 1 SLICE 1 SLICE 1 SLICE 1 BIMASC 1 RASP SYCOT SYCOT 1

o

Construction of the Hankelmatrix expansion of a d multivariable parameter sequence

HANKEX SYCOT

o

c Computes a Toeplitz matrix expansion of a time d sequence at a Tilled moment

Computes UU , with U the Toeplitz matrix d expansion of a given time sequence

CHANKX TOEPEX UUTR SYCOT SYCOT SYCOT 1

o

1

(40)

-

39-b. Trans/or11U1J;ion routines

Monovariable impulse response sequences H(n.a.1/> •... ) for a pole of multiplicity n. damping a and sample angle

I/>

Multivariable impulse response of a given ssr Markov parameters of a multivariable ARMA-model

Markov parameters of a given ssr

Impulse response from input/output data using the correlation method

Multivariable impulse response by deconvolution 3.9. Balancing transformations

Balances the subsystem of a time-invariant ss-model

Reduces a given ss-representation to numerically balanced form

Computes the balancing transformation Balances a linear ss-model

Backtransformation of a balanced system

Balances (interactively) a not necessarily minimal ss-system

Computes the balancing transformation of lABLNC t d c d d d d d d r d d d d r r name source i GENMAR SYCQT 1 CGENMR SYCOT 1 MARKOV SYCOT 0 ARMAH SYCOT 0 MARKOV RASP 1 HCORR SYCOT 1 lOrn SYCOT 2 BALRNM BIMASC 1 SSBAL SLICE 1 TSBAL BIMASC 1 TSBALI BIMASC 1 EQUIL RASP 1 REQUIL RASP 1 lABLNC SYCOT 1 BLNC SYCOT 1

(41)

4. ANALYSIS ROUTINES

t name source i

4.1. State Space (SS) and Generalized State Space (GS)

4.1.0. Auxiliary routines

Annulates small elements in the matrices of a d NULL RASP 1

ss-model

Elementary transformation with permutation d PTRA RASP 1 of a linear system

Elementary orthogonal transformation of a d HOUS RASP 1 linear system

Permutation of states d UMORD RASP 1

4.1.1. Canonical and quasi-canonical forms

Standard controllability form of a single input d TMICOl BlMASC 1 linear ss-system by stabilized elementary

simi-larity transformations

Transformation matrix for (upper )staircase d DSTAIR SYCOT 0 form

Constructs a minimal order ssr in observable d REMOI BIMASC 1 canonical form

Constructs a minimal order ssr in controllable d REMCI BIMASC 1 canonical form

Reduction of a single input system into canoni- r TRSCF LISPACK 1 cal form by orthogonal transformation

Reduction of a multi input system into canoni- r TRMCF LISPACK 1 cal form by orthogonal transformation

FN-form of a ssr given in HN-form d FROB RASP 1

HN-form of a given ssr d ELDIN RASP 1

Lower Hessenberg form of a single input sys- r HESCYC AUTLIB 1 tem

Lower Hessenberg form of a multi-input sys- r MULHES AUTLIB 1 tern

4.1.2. Change of basis

Applies a general orthogonal system similarity d ORTEQ BIMASC 1 transformation to a state space description

Applies a general system similarity transfo'r- d SIMEQ BIMASC 1 mation to a state space description

4.1.3. Structural indices

Controllability. observability or decouplingsin- d SBEIND RASP 1 dices

Decouplingsindices of a system in HN-form d ENTKOP RASP 1

(42)

-

41-4.1.4. Continuous/discrete time

Computes the sampled data system corresponding to a given continuous ssr

Discrete time from continuous time ss-model Discrete time from continuous time ss-model. or vice versa

Solution of the continuous time ss-equations Solution of the discrete time ss-equations 4.1.5. Interconnection of subsystems

4.1.6. Controllability, observability Reachable or unobservable subspace Controllable subspace

Controllability/observability matrix of a linear system

Controllable or observable part via HN-form Controllability and observability test of a ss-model

Controllability test of a given ss-model Observability test of a given ss-model

Determination of a linearly independent vector of a controllability matrix

4.1.7. Inverse systems 4.1.8. Poles, zeros, gain

11. Poles and zeros of ass-model

Pole-zero map of a multivariable system Poles and residues of ass-model

Computes the invariant zeros of ass-model

Determines a reduced system of a ssr having the same system zeros

21. Gain (SISO) of a transfer function from its ssr Computes the steady state gains (MIMO) for a given transfer function matrix

4.1.9. Model reduction

Evaluation of dominant states and eigenvalues Dominance measures

Model reduction of a modal transformed sys-tem by dominant mode analysis

t d d d r d d d d r d d d d r d d d r r d d d d d d d d name source i TSCD BIMASC 1 ABTAST RASP 1 SSTRAN SYCOT 2 BILNTR SYCOT 1 RUN SYCOT 2 RUNDIS SYCOT 2 DSTAIR SYCOT 0 DDEADB SYCOT 0 CONMAT AUTLIB 1 REDHN RASP 1 CONOBS RASP 1 CONTRL SYCOT 1 OBSER SYCOT 1 COOBIN AUTLIB 1 EIGSYS SYCOT 1 EIGVA SYCOT 1 TFPART RASP 1 SSZER SLICE 1 SSTZER SLICE 1 MZEROS BIMASC 1

ZERO. REDUCE SYCOT 1

MREDUC BIMASC 1

GAIN. GAIN 1 BIMASC 1

MTVAR BIMASC 1

WEZU RASP 1

DOMWES RASP 1

(43)

t Dominance analysis of the eigenvalues of a d modal transformed system

Optimal Hankelnorm approximant of a bal- r anced continuous ss-system

4.1.10. -Almost-(~B) invariant subspaces 4.1.11. -Almost- controllability subspaces 4.1.12. Scalar and multivariable root loci 4.1.13. Nyquist diagrams 4.1.14. Bode diagrams name MODOM GLOVER Bode diagrams

Logarithmic frequency response of ass-model

BOPLOT rId BOLAUB 4.1.15. Simulation

Solution of a continuous time invariant linear d system with transitionmatrix

System response of a continuous linear system d Step response of a continuous or discrete linear d system

Evaluates y == Cxx

+

Dxu. where y, x and u d are the system output. state and input vectors. respectively

Evaluates A xx + B xu

+

BZ xw. where x. u d and ware the system state. control and distur-bance vectors. respectively. and A is a square matrix in upper Hessenberg form

Evaluates the right-hand side of the linear sys- d tem of differential or difference equations corresponding to various multivariable control structures

Simulation of the control system d

INTEAT SYSAT SPRANT OUTP STATE GSTEP SYSTEM source RASP SYCOT RASP RASP RASP RASP RASP BIMASC BIMASC BIMASC TIMSAC i 1 1 1 1 1 1 1 1 1 1 1

(44)

-

43-t name source i

4.2. Polynomial Matrix Analysis (PM) 4.2.1. Canonical and quasi canonical forms 4.2.2. Equivalence transformations

4.2.3. Greatest common divisor

Greatest right divisor of a rectangular pm ? RDIV KONTOS 2 Greatest common divisor of two pm's ? GCRD SYCOT 2 4.2.4. Continuous/discrete time

4.2.5. Interconnection of subsystems 4.2.6. Controllability, observability 4.2.7. Inverse systems

4.2.8. Poles, zeros

All zeros of a polynomial matrix ? ZPOLM KONTOS 2 Right divisor of a pm with zeros in a given ? SPFE KONTOS 2 region

4.2.9. Model reduction 4.2.10. Root loci

4.2.11. Nyquist diagrams

Nyquist diagrams of scalar transfer functions NYPLOT RASP 1

of discrete or continuous systems 4.2.12. Bode diagrams

Gain and phase in a given interval (1og) from a rid FRELOG RASP 1

scalar factorized transfer function, discrete or continuous

(45)

t name :>Uu~vc

iJ

4.3. Rational Matrix Analysis (RM) 4.3.1. Equivalence transformations 4.3.2. Structural indices

4.3.3. Continuous! discrete time 4.3.4. Interconnection of subsystems 4.3.5. Inverse systems

4.3.6. Poles~ zeros

Poles and zeros of a transfer matrix rid TFMRP RASP 1

4.3.7. Model reduction 4.3.8. Root loci

Number of cuts of a root locus with a ree- r AESTE RASP 1 tangular frame

Angular function evaluation for root loci r WIFU1 RASP 1

d WIFUD RASP 1

Root loci points d WOKPU1 RASP 1

Root loci curves d WOK 1 RASP 1

Drawing of root loci r WOPLOT RASP 1

Configuration of poles and zeros for root loci d KONFIG RASP 1

Plots the pole-zero configuration in a root loci KRIPLO RASP 1 diagram

4.3.9. Nyquist diagrams

Calculates a Nyquist diagram from a rational. r SSFRNY SYCOT 1 continuous transfer function

Computes a complete Nyquist plot d PUNKTE RASP 1 Computes complete Nyquist. Popov. Tsypkin r ORTFUN RASP 1 plots for a rational transfer function

Draws Nyquist and Popov plots and frequency d NYPLOT RASP 1 loci for z-transformations

4.3.10. Bode diagrams

Calculates a Bode diagram from a rational, r SSFRBD SYCOT continuous transfer function

(46)

-

45-t name source i

4.4.

Frequency Response Analysis (FR.)

4.4.1.

Polar/rectangular coordinates

4.4.2.

Interpolation

4.4.3.

Inverse systems

4.4.4.

4.4.5.

4.5.

Time Response Analysis (TR)

4.5.1.

Scaling

4.5.2.

Interpolation

4.5.3.

Convolutio~ deconvolution

4.5.4.

4.6.

Markov Parameter Analysis (Mp)

4.6.1.

Scaling

4.6.2.

Interpolation

4.6.3.

Convolutio~ deconvolution

4.6.4.

4.6.5.

Controllability, observability

4.6.6.

Change of basis

4.6.7.

Model reduction

4.7.

Stability

Mansour stability test for linear continuous r MANSTB AUTLIB 1

(47)

5. SYNTHFSIS ROUTINES

t

5.1. State Space Synthesis (SS)

5.1.1. Eigenvalue/eigenvector assignment

a. Assignment

Pole assignment for a single input system r

Pole assignment synthesis r

Pole placement by state feedback r Pole assignment by state feedback using the d Schur method

Feedback vector for pole placement d Feedback gain for eigenvalue assignment d Pole assignment for a single input system by r use of a Hessenberg algorithm

b. Stabilization

Computes a stabilizing gain matrix. continuous d system

d

Computes a stabilizing gain matrix. discrete d system

d

5.1.2. Riccati equations

11. Steady state, continuous/discrete

a. Continuous

Constructs the Hamiltonian matrix for solving d CARE

Solution of the continuous algebraic matrix d Riccati equation (CARE) (Laub), and optimal steady state feedback gain

Solution of CARE (Laub's Schur form method) r

d

Solution of CARE (Kleinman) with stabiliza- d tion according to Armstrong

Solution of CARE with stability margin assign- d

ment

Solution of CARE d

Solution of CARE with matrix sign function d

Solution of CARE (Newton) d

Solution of CARE (iterative Newton) d Time invariant CARE with an eigenvalue r method

Residual of an approximate Riccati solution d Optimal control via the matrix Riccati equation r (continuous systems) name SIPASS POLSC POLSC SALOC POSlHE KPOL POSIHE STAC CSTAB STAD DSTAB EXTC KRICLB RILAC XRICCA KRICNT RICAT KRINWT ROBERT NEWTON NTNC LCRFBI RESID SOLCS. RICSL source

I

iJ

AUTLIB 1 LISPACK 1 SLICE 1 BIMASC 1 RASP 1 RASP 1 AUTLIB 1 BIMASC 1 RASP BIMASC RASP BIMAS 1 1 1 1 RASP 1 SLICE 1 SYCOT 0 RASP 1 RASP 1 RASP 1 BYERS 1 BYERS 1 BIMAS 1 AUTLIB 1 BYERS 1 LISPACK 1

(48)

-

47-b. Disa-ete

Constructs the extended symplectic matrix for solving DARE

Solution of DARE (iterative Newton)

Solution of DARE (Laub) and optimal steady state feedback gain

Solution of DARE (Kleinman) with stability margin assignment

Solution of DARE (Kleinman)

Time invariant Riccati feedback matrix of a discrete time system

Hamiltonian matrix for solving the Riccati equation

Optimal control via the matrix Riccati equation (discrete systems)

State space optimal regulator gain of DARE

c. Continuous or discrete

Constructs the matrices defining the general-ized eigenvalue prOblem for solving the (near) singular CAREIDARE

Schur vector method (Laub)

Generalized Schur method (Van Dooren) Generalized Hamilton method (Van Dooren)

21. Time varying

Finite interval discrete optimal control (dual to Kalman-fi.l ter )

5.1.3. Lyapunovequations

a. Continuaus

Solution of ATX

+

XA

=

C. A quasi-triangular. C symmetric

Idem. A arbitrary or Schur form. C symmetric

Idem. A arbitrary. C symmetric

Idem. A arbitrary. C factorized as BTB Idem. A upper Schur form. C symmetric

b. Disa-ete Solution ofAXC + B

=

X Solution of AT XA - X

=

C. C symmetric t d d d d d r rId r d d d d d d d d d d d rid d r d d r r d d r d r name source i EXTD BIMAS 1 NTND BIMAS 1 DRICLB RASP

1

DRICNT RASP 1 DRINWT RASP 1 LDRFBI AUTLIB 1 AUD RASP 1

SOLDS. RIDSL LISPACK 1

XDRICC SYCOT 0 EXT2 BlMAS 1 SCHV BIMAS 1 GSCHV BIMAS 1 DXTHAM SYCOT 0 SQUAR1 SYCOT 1 SQUAR2 SYCOT 1 SRCF SYCOT 0 BCKSLV BYERS 1 SYMSLV RASP 1 SYMSLV SYCOT 0 ATXPXA RASP 1 ATXPXA SYCOT 0 LYBSC SLICE 1 LYAPUN BYERS 1 CLYA SYCOT 1 LYCSL.LYCSR LISPACK 1 SPDLY SLICE 1 LYAC BIMAS 1 SUM RASP 1 LYBAD SLICE 1 FXFTPS SYCOT 0