an inventory of basic software for computer aided control

An inventory of basic software for computer aided controlsystem 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

May 1986

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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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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. consequently. 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 available. Some packages are free. Others are free. or available against a nominal fee. for educational 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 problemoriented. 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. Comments 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 81/3. November 1981.

Library index

1. UTILITY ROUTINES CUT) 1)

1.1. Text handling

1.2. File handling

1.3. Graphical input/output

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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

Other unconstrained problems

Linearly constrained linear least squares 2.3.7. Linearly constrained nonlinear least squares 2.3.8. Other linearly constrained problems

2.3.9. Nonlinearly constrained nonlinear least squares

2.3.10. Other nonlinearly constrained problems

1) The letters within brackets in the heading of a (sub)section have to do with the naming convention proposed in the SYCOT report on implementation and documentation standards.

2.4. Zeros and nonlinear equations

2.4.1. Zeros of a polynomial

2.4.2. Zeroes) of a function

-5-

2.4.3. Systems of nonlinear equations

2.5. Differential equations

2.5.1. Initial value problems

2.5.2. Boundary value problems

2.5.3. Partial differential equations

3. TRANSFORMATION ROUTINES

3.1. State space

3.2. Generalized state space 3.3. Polynomial matrix fractions 3.4. Polynomial matrix quadruples

3.5. Rational transfer functions

3.6. Frequency response

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

3.8. Markov parameters

3.9. Balancing transformations

4. ANALYSIS ROUTINES

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

4.1.0. Auxiliary routines

4.1.1.

4.1.2.

4.1.3.

4.1.4.

4.1.5.

4.1.6.

4.1.7.

4.1.8.

Canonical and quasi canonical forms

Change of basis

Structural indices

Continuous/discrete time

Interconnection of subsystems

Controllability. observability

Inverse systems

Poles, zeros, gain

4.1.9. Model reduction

4.1.10. (A, B) invariant and almost (A, B) invariant subspaces

4.1.11. Controllability and almost controllability subspaces

4.1.12. Scalar and multivariable root loci

4.1.13. Nyquist diagrams

4.1.14. Bode diagrams

4.1.15. Simulation

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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


4.2.10. Root loci


4.2.12. Bode diagrams 4.3. Rational Matrix Analysis CRM)


4.3.2. Structural indices

4.3.3. Continuous/discrete time 4.3.4. Interconnection of subsystems


4.3.6. Poles. zeros


4.3.8. Root loci


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. Continuous/discrete time 4.4.5. Interconnection of subsystems

4.5. Time Response Analysis (TR)

4.5.1. Scaling 4.5.2. Interpolation 4.5.3. Convolution. deconvolution 4.5.4. Interconnection of subsystems

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4.6. Markov Parameter Analysis (MP)

4.6.1. Scaling

4.6.2. Interpolation

4.6.3. Convolution. deconvolution


4.6.5. Controllability. observability

4.6.6. Change of basis


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)

-8-

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


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. 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 Institute for Management and Informatics. Bucharest. 1982. A.Varga, V.sima. BIMAS - A Basic Mathematical Package for Computer Aided Systems 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 Subprograms 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. Ph. D. Thesis. Dept. Compo Sc., Cornell University. 1983.

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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 Verlag, Berlin. Heidelberg. New York, 1977.

8. EBLAS

An Extension to the Set of Basic Linear Algebra Subprograms. targeted at matrix vector 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 Publications. Philadelphia. 1979.

- 12-

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 ComputerAided 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 equations.

L.it. A.Hindmarsh. ODEPACK: a systematized collection of ODE solvers. in Scientific Computing: Applications of Mathematics and Computing to the Physical Sciences. R.S.Stepleman. Ed. (IMACS Transactions on Scientific Computation. 10th IMACS World Congress. Montreal 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. Control Systems Research Group. Internal report 81/3. November 1981.

- 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 monographs. 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

4: a method described in literature

-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 precision 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

-16 -

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

- 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

-18 -

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 precision 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

- 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

- 20-

t name source i Idem. A Hermitian band matrix

I ~ CHBMV BBLAS 4 ZHBMV BBLAS 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 z ZTBMV BBLAS 4

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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

- 22-

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

- 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 z ZTRCO UNPACK 1

- 24-

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 tridiagonal 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-

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

- 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

SEPEST BYERS 1

- 28-

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 transformation

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

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

- 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 GCHOL SYCOT 0

- 30-

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

- 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 approximations 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-

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

- 34-

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

- 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

- 36-

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 equations

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

- 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

- 38-

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 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

- 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 ARMAmodel 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 ssmodel 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

- 40-

4. ANALYSIS ROUTINES

t name source i 4.1. State Space (SS) and Generalized State Space

(GS)


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


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


Controllability. observability or decouplingsin- d SBEIND RASP 1 dices Decouplingsindices of a system in HN-form d ENTKOP RASP 1 Controllability and Kronecker indices d INDEX RASP 1

- 41-


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


4.1.6. 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 ssmodel Controllability test of a given ss-model Observability test of a given ss-model Determination of a linearly independent vector of a controllability matrix


4.1.8. 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


Evaluation of dominant states and eigenvalues Dominance measures Model reduction of a modal transformed system 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 SYSRED RASP 1

- 42-

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



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 disturbance 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

- 43-

t name source i 4.2. Polynomial Matrix Analysis (PM)

4.2.1. Canonical and quasi canonical forms


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





4.2.8. 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


4.2.10. Root loci


Nyquist diagrams of scalar transfer functions NYPLOT RASP 1 of discrete or continuous systems


Gain and phase in a given interval (1og) from a rid FRELOG RASP 1 scalar factorized transfer function, discrete or continuous Plots a Bode diagram BOPLOT RASP 1

- 44-

t name :>Uu~vc iJ 4.3. Rational Matrix Analysis (RM)



4.3.3. Continuous! discrete time



4.3.6. Poles~ zeros

Poles and zeros of a transfer matrix rid TFMRP RASP 1


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


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


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

- 45-

t name source i 4.4. Frequency Response Analysis (FR.)

4.4.1. Polar/rectangular coordinates





4.5. Time Response Analysis (TR)

4.5.1. Scaling


4.5.3. Convolutio~ deconvolution


4.6. Markov Parameter Analysis (Mp)

4.6.1. Scaling


4.6.3. Convolutio~ deconvolution





4.7. Stability

Mansour stability test for linear continuous r MANSTB AUTLIB 1 time systems

46

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

- 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 generalized 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 L YDSL. LYDSR LISPACK 1

- 48-

t name source i Solution of ATXA + C = X. C symmetric. A d LYAD BIMAS 1 upper Schur form Solution of AXAT + C = X. C symmetric. A d SYMSOL SYCOT 0 lower Schur form Solution of the discrete time Lyapunov equa- d DLYA SYCOT 1 tion

c. Continuous or discrete Observability and controllability Gramians of r LYAP SYCOT 1 ass-system

d. Generalized Lyapunov equation Solution of AX + XB + C - O. A and B negative r AXXBC AUTLII3 1 definite. C arbitrary

S.IA. Sylvester equations

Solution of AX + XB - C. (Hessenberg Schur method)

r SYHSC SLICE 1

d SYLHC BIMAS 1 Idem. A lower. B upper quasi-triangular d SYLSC BIMAS 1 Idem. A lower. B upper Schur form. with exit d SYLSM BIMAS 1 if the magnitude of any solution element exceeds a given bound Idem. A lower. B upper Schur form r SHRSLV SYCOT 0 Idem. nothing known about A. B. C r SYCSL.SYCSR LISPACK 1

Solution of AXB + X-C. (Hessenberg Schur method)

d SYLHD BIMAS 1

Idem. A lower. B upper quasi-triangular d SYLSD BIMAS 1 Solution of AP + PB ... - Q. A and B arbitrary d SMITH RASP 1 but stable


5.1.6. Dead beat control

Dead beat control d DDEADB SYCOT 0 Generalized dead beat control ? GDEADBEAT SYCOT 2

5.1.7. Observers

Minimal order state estimator for a time- d SAESTM BIMASC 1 invariant continuous or discrete system Pole assignment to Luenberger observer of r NOOP AUTLIB 1 order n Discrete time n-th order Luenberger observer r NOLOBS AUTLIB 1 Parameters of the functional observer for a r CONOBS AUTLIB 1 SIMOsystem

5.1.8. Spectral factorization

- 49-

t name source i 5.1.9. Realization methods

Minimal state space model from a system given d HREAL SYCOT 1 by its Markov sequence Accuracy check of the model obtained by reali- d HDIFF SYCOT 1 zation

5.1.10. Optimal regulator problems

Linear optimal sampled data regulator from d SAMDA RASP 1 linear optimal continuous regulator Elimination of cross-product term in the qua- d PREFIL RASP 1 dratic performance index Weighting matrices for the linear optimal sam- d DIWI RASP 1 pled data Optimal state feedback gain matrix from the d OPTR BIMASC 1 solution of a CAREIDARE Optimal control of a linear continuous system r NU.NUZ AUTLIB 1 Dynamical compensator of a controllable SIMO r COMPA AUf LIB 1 system Optimal dynamical compensator of a controll- r COMRIC AUTLm 1 able SIMO system Solution of the linear regulator problem and r OPTIFI AUf LIB 1 calculation of the prefilter gain Regulator synthesis of a nonlinear continuous r OPTC AUTLm 1 system by unconstrained optimization Optimal control from the gain matrices d CONTRL TIMSAC 1 Optimal controller gain matrices d OPTDES TIMSAC 1

5.1.11. Hierarchical control

5.1.12. Decentralized control

Tests if a multi input system has property D r DSIMO AUTLIB 1


5.1.14. Model matching

Computes the disturbance and reference feed- d STFF BIMASC 1 forward gain matrices for a time-invariant sys-tem described by ass-model Computes the disturbance and reference feed- d MTFF BIMASC 1 forward gain matrices for a time-invariant sys-tem described by a transfer function matrix

- 50

t name source i 5.2. Polynomial Matrix Fraction Synthesis (PM)

5.2.1. Eigenvalue! eigenvector assignment



5.2.4. Model matChing

5.2.5. Parameter optimization

5.3. Rational Matrix Models Synthesis (RM)

SA. Frequency Response Models Synthesis (FR)

SS. Time Response Models Synthesis (TR)

Connects control loop and dynamic regulator rId GESYSO RASP 1 into a total system System-expansion with a homogeneous input- rId HOMGA2 RASP 1 signal generator

5.6. Markov Parameter Models Synthesis (Mp)

- 51-

6. DATA ANALYSIS (DA.)

t name source i 6.1. Scaling, interpolation

Decimation. interpolation or filtering of a sig- r DIFILT DSP 1 nal Conversion of the sampling rate of a signal by r SRCONV DSP 1 the ratio LIM Optimal digital interpolating filter d DODIF DSP 1

6.2. Statistical properties

White noise variance and (-2)log likelihood of d FUNCT2 TIMSAC 1 a data sequence

63. Trend removal

Removes the DC component and slope of a sig- r LREMV DSP 1 nal

6.4. Covariances

Correlation coeffients between two multivari- d ACCOR SYCOT 0 able sequences Convolution and deconvolution product r FCD SYCOT 0 Autocorrelation function of a signal r FAC SYCOT 0 Crosscorrelation function of two signals r FCC SYCOT 0

6.5. Spectra

Sine and cosine transforms r FSC SYCOT 0 Powerdensity spectrum r PSD SYCOT 0 Power spectrum of an ARMA process d NRASPE TIMSAC 1 Real cepstrum of a real sequence r RCEPS DSP 1 The inverse complex cepstrum r ICCEPS DSP 1 Complex cepstrum of a sequence r CCEPS DSP 1

6.6. Discrete Fourier transforms

11. Onedimensional - one channel Cooley-Tukey fast Fourier transform c FOUREA DSP 1 Finite discrete Fourier transform (DFT) of a r FAST DSP 1 real vector Fourier synthesis of a real vector from the r FSST DSP 1 Fourier coefficients Finite DFT for a real vector (radix 8 algo- r FFA DSP 1 rithm) Fourier synthesis for a real vector from the r FFS DSP 1 Fourier coefficients (radix 8 algorithm) Finite DFT for complex data r FFT842 DSP 1 DFT for a real. symmetric. N-point sequence r FFTSYM DSP 1

- 52-

t

Inverse discrete Fourier transform (IDFT) for a r real, symmetric. N-point sequence DFT for a real. anti-symmetric. N-point r sequence IDFT for a real. anti-symmetric. N-point r sequence DFT for a real, odd harmonic. N-point r sequence IDFT for a real. odd harmonic. N-point l'

sequence DFT for a real, symmetric. odd harmonic. N- r point sequence IDFT for a real, symmetric. odd harmonic. N- r point sequence DFT for a real, anti-symmetric. odd harmonic. r N-point sequence IDFT for a real. anti-symmetric. odd harmonic. l'

N-point sequence Radix 2 fast Fourier transform r Mixed radix fast Fourier transform r

r Mixed radix fast Fourier transform (complex r signal) Winograd Fourier transform l'

Time-efficient forward or inverse complex DFT l'

via radix 4 FFT Fourier transform (Goertzel method) d

12. Onedimensional - multic1u:t.nnel Multivariate complex Fourier transform. using l'

mixed radix algorithm With FFT to compute Fourier transform or r inverse for real data With FFT to compute Fourier transform or r inverse for real data. single- or multivariate

21. Multidimensional Optimized multidimensioned mass storage FFT' r real to complex or vice-versa Optimized mass storage complex FFT c Two-dim. FFT for real/complex data r Two-dim. IFFT for real/complex data r

6.7. Z-transforms

llame IFTSYM

FFTASM

IFTASM

FFTOHM

IFTOHM

FFTSOH

IFTSOH

FFTAOH

IFTAOH

FFT RLTR FFTMX FFT

WFTA RADIX4

FOUGER

FFT

REALS

REALT

RMFFT

CMFFT FFT2T FFT2I

CHIRP Z-transform r CZT

6.8. ~ictioll

01. Auxiliary rautines Covariance matrix of a given signal r COVAR1

source i DSP 1

DSP 1

DSP 1

DSP 1

DSP 1

DSP 1

DSP 1

DSP 1

DSP 1

LPS 3 SYCOT 0 DSP 1 SYCOT 0

DSP 1 DSP 1

TIMSAC 1

DSP

DSP

DSP

DSP

DSP DSP DSP

DSP

DSP

1

1

1

1

1 1 1

1

1

- 53-

t (M+ 1) x (M+ 1) covariance matrix using the M r x M covariance matrix and the signal Transformations between various parameter r sets used in linear prediction

11. Correlation. metlwds

name COVAR2

LPTRN

Linear prediction analysis using the autocorre- r AUTO lation method

21. Covariance metlwds

source i DSP 1

DSP 1

DSP 1

Linear prediction analysis using the covariance r COY AR DSP 1 method Square root covariance filter r FILT1. FILT2 AUTLIB 1

22. Lattice algorithms General covariance lattice algorithm for linear r COVLAT prediction Linear prediction by a covariance lattice rou- r CLHARM tine for harmonic mean method

6.9. Windowing

Data windowing of a correlation function: han- r ning window, hamming window, quadratic window Triangular window r Generalized Hamming window r Kaiser window r Chebyshev window parameters r Dolph Chebyshev window design r Operates as a data window d

6.10. Filter design

a. Finite impulse response design

LDW

TRIANG HAMMIN KAISER CHEBC CHEBY WINDOW

Filters one frame of data for a given filter fre- r RFILT quency response Remez exchange algorithm for the weighted r REMFZ Chebyshev approximation of a continuous function with a sum of cosines Coefficients of a maximally flat FIR linear r MXFLAT phase filter with odd number of terms and even symmetry in filter coefficients Design of linear phase FIR-filters in direct r IDEFIR form with minimum coefficient word length

b. Infinite impulse response design

DSP 1

DSP 1

SYCOT 0

DSP 1 DSP 1 DSP 1 DSP 1 DSP 1 TIMSAC 1

DSP 1

DSP 1

DSP 1

DSP 1

- 55-

t name source i e. Variance matrices

Residual variance of a subset regression model d COMPSD TIMSAC 1 . Residual variance of a regression model d SDCOMP TIMSAC 1 Subset regression coefficients and residual vari- d SRCOEF TIMSAC 1 ance computation Variance matrix of a stationary state vector by d SUBPM TIMSAC 1 the procedure of Akaike

7.2.1. Covariance methods

Correlation least squares method r CLS SYCOT 0 Fits an ARMA model to stationary scalar time d CANCOR TIMMC 1 series Future canonical weights and the order of the d CANOCO TIMSAC 1 Markovian model

7.2.2. Deconvolution, numerical normal equations

7.2.3. Bayes estimation

Partial autocorrelation coefficients of the Baye- d BAYSPC TIMSAC 1 sian model Bayesian weight of the AR model of each order d BAYSWT TIMSAC 1 Bayesian procedure with models of succes- d ARBAYS TIMSAC 1 sively increasing order Bayesian type non-stationary AR-model fitting d NONSTB TIMSAC 1 procedure Bayesian estimates of partial correlations by d SUBSPC TIMSAC 1 checking all subset regression models Bayesian model based on all subset regression d SBBAYS TIMSAC 1 models Partial AR coefficients of the multivariate AR d MBYSPC TIMSAC 1 model Multivariate AR model fitting by a Bayesian d MBYSAR TIMSAC 1 procedure Multivariate AR model fitting to instationary d MNONSB TIMSAC 1 time series by a Bayesian procedure

7.2.4. Maximum likelihood

Maximum likelihood iteration of an ARMA r MLH SYCOT 0 model for the process and a MA model for the noise Estimation of the continuous-time parameters r SVM SYCOT 0 of a state space model Inverse of an approximation to the Hessian of a d HESIAN TIMSAC 1 log-likelihood function of the AR model of order k Minimum AIC procedure with AR models of d ARMPIT TIMSAC 1 successively increasing order Controls the maximum likelihood computation d SMINOP TIMSAC 1

- 56-

t name Exact maximum likelihood estimates of the d ARMLE parameters of an AR model Exact likelihood and its gradient of the m-th d FUNCT order AR model Multivariate AR model fitting using the d MARFIT minimum AIC procedure Minimum AIC type subset regression analysis d SUBSET

7.2.5. Least squares methods

Least squares estimation using pseudo inverse r Least squares estimation using orthogonal r functions Generalized least squares. low order noise. r iterative technique. ARMA model for process. AR model for noise Generalized least squares. high order noise. r iterative technique. ARMA model for process. AR model for noise Generalized least squares. recursive technique. r ARMA model for process, AR model for noise Extended least squares iteration. ARMA model r for process. MA model for noise Recursive techniques for ARMA process and r MA noise: simple least squares. extended least squares. least squares with instrumental variable Least squares finite impulse response system c identification Direct LIP-identification of a linear discrete r time SISO system Least squares estimates of partial AR coeffi1i.ent d matrices of a multidimensional AR model

7.2.6. Instrumental variable methods

7.2.7. Model reference methods

7.2.8. Prediction error methods

7.2.9. Stochastic approximation

7.2.10. Order/structure determination

LSA LSB

GLA

GLB

REC

ELS

RECUR

FIR

DLIP

MPARCO

Product moment (determinant ratio) test Instrumental product moment test

r ORC r OR!

source i TIMSAC 1

TIMSAC 1

TIMSAC 1

TIMSAC 1

SYCOT 0 SYCOT 0

SYCOT 0

SYCOT 0

SYCOT 0

SYCOT 1

SYCOT 1

SYCOT 3

AUTLIB 1

TIMSAC 1

SYCOT SYCOT

o o

- 57-

t name source i 7.3. General methods

7.3.1. Parameter and state estimation combined

Off line identification of a discrete transfer r ID1D AutLIB 1 function Identification of a (non)linear MIMO system r NLID AUTLIB 1 using an output error method

7.3.2. Use of deterministic signals

7.3.3. Evaluation of input signals

7.3.4. Test of model structure

Checks the stability of the AR or MA part of a d ARCHEK TIMSAC 1 model

- 58-

8. FILTER THEORY (FT)

t 8.1. Kalman filters

a.

b.

Conventional Kalman filters Solution of a matrix Riccati difference equation for discrete Kalman filter and Kalman gain Recursive Kalman filtering State estimation of a time invariant system in the steady state case Steady state discrete time Kalman Bucy transfer matrices Discrete time stabilized Kalman Bucy filter for systems with nonzero system noise

Square root Kalman filters Chandrasekhar Covariance, Hessenberg form Covariance, Schur form Information, Schur form

rid

d r

r

r

d d d d

8.2. LPC filters

Linear predictor polynomials and reflection r coefficients by autocorrelation method Linear predictor polynomials and reflection r coefficients by covariance method Predictor polynomials from reflection r coefficients Reflection coefficients from predictor polynomi- r als Transformation of a rational function to r reflection coefficients and tap weights Output of a lattice filter applied to a time r series

r

name

SAMPL

RKF DSSKF

DSSKFM

FKALSU

CSRF SQUAR1 SQUAR2 SRIF

source

RASP

SYCOT AUTLIB

AUTLIB

AUTLIB

SYCOT SYCOT SYCOT SYCOT

AUTO!, AUT02 LPS

COVAR!, COY AR2 LPS

STEPUP LPS

STEPDN LPS

EVAL LPS

DIRECT, TWOMUL LPS

KLOCH, ONEMUL LPS

(

(

(

(

(

- 59-

Alphabetic index

The next pages contain the routines from the inventory arranged in alphabetic order per source (library/package). It gives an overview of the routines of the source which are included. and facilitates the search for the place in the inventory of a specific routine. For each routine the number of the section(s) in which it occurs and a short description is given. Not included are the packages with only mathematical routines: BLAS. EISPACK. UNPACK. MINPACK. ODEPACK. SSP. The documentation of each of these packages (cf. the references in section Sources. p. 10 if) is generally accessible. This documentation gives complete information of the contents of the package in question.

AUTLIB

name ABCS AXXBC

COMPA COMRIC CONMAT CONOBS COOBIN

DEMIHE

DLIP DSIMO DSSKF

DSSKFM EULER FILT1 FILT2 FKALSU

GCON GOLSEC GUNC HESCYC HESPOL HESSCO

HHUNIT HOTRAN HOUTRA IDID INTSUC LCRFBI LDRFB1

LINMIN LSP2 MANSTB MULHES MXQXT MXTQX NLID

NLP NOLOBS NOOP NORM NRELDI

NU

section 2. 1. 1.22.b 5.1.3.d

5.1.10. 5.1.10. 4.1.6. 5.1.7. 4.1.6.

2.1.3.03.

7.2.5. 5.1.12. 8.1.a

8.1.a 2.5.1.11. 6.8.21. 6.8.21. 8.1.a

2.3.1.11. 2.3.5.11. 2.3.1.11. 4.1.1. 2.1.3.03. 2.1.3.02.c

2.1.1.21.e 2.1.4.21. 2.1.4.21. 7.3.1. 2.3.5.11. 5.1.2.11.a 5.1.2.11.b

2.3.2. 2.1.5.11. 4.7. 4.1.1. 2.1.1.21.b 2.1.1.21.b 7.3.1.

2.3.10. 5.1.7. 5.1.7. 2.1.1.21.c 3.1.31.

5.1.10.

- 60-

description Product: AB with A and B such that AB is symmetric Solution of AX + XB + C = O. A and B negative definite. C arbitrary Dynamical compensator of a controllable SIMO system Optimal dynamical compensator of a controllable SIMO system Controllability/observability matrix of a linear system Parameters of the functional observer for a SIMO system Determination of a linearly independent vector of a controllability matrix Characteristic polynomial and cofactors of the last row elements of sI - A. for a lower Hessenberg matrix A Direct LIP-identification of a linear discrete time SISO system Tests if a multi input system has property D State estimation of a time invariant system in the steady state case Steady state discrete time Kalman Bucy transfer matrices Solution of a system of first order differential equations Square root covariance filter Square root covariance filter Discrete time stabilized Kalman Buey filter for systems with nonzero system noise Evaluates the gradient of a function and restrictions Minimum of a scalar function in a predetermined interval Evaluates the gradient of a function Lower Hessenberg form of a single input system Characteristic polynomial of a lower Hessenberg matrix Transformation of an arbitrary (sub)matrix to lower Hessenberg form Initialization of a matrix by a unity matrix Transformation of a matrix into a triangular matrix QR factorization with column permutation Off line identification of a discrete transfer function Minimum of a scalar function Time invariant CARE with an eigenvalue method Time invariant Riccati feedback matrix of a discrete time system Minimal LLS-solution. arbitrary matrix Computes the matrix exponential and its integral Mansour stability test for linear continuous time systems Lower Hessenbe? form of a multi-input system Product of XQX • X arbitrary. Q symmetric Product of XT QX. X arbitrary. Q symmetric Identification of a (non)linear MIMO system using an output error method Nonlinear mathematical programming problem Discrete time n-th order Luenberger observer Pole assignment to Luenberger observer of order n Frobenius norm of a square matrix Minimal realization of a transfer matrix by the method of Nour-Eldin Optimal control of a linear continuous system

name NULL NUZ OPTl OPTC

OPTIFl

PHAVA

POLAR POSIHE

RAND RK RKS8 RUNU

SIPASS SMADD SMORTH SORTAG UNCOP

UNIOP

88Ction 2.1.1.22.e 5.1.10. 2.3.5.11. 5.1.10.

5.1.10.

3.1.41.

2.0.1. S.l.l.a

2.0.4. 2.5.1.11. 2.5.1.11. 2.5.1.11.

S.1.1.a 2.1.1.22.a 2.1.4.21. 2.0.3. 2.3.5.12.

2.3.1.21.

- 61-

Annulates a part of a matrix Optimal control of a linear continuous system Minimum of a scalar function Regulator synthesis of a nonlinear continuous system. by unconstrained optimization Solution of the linear regulator problem and calculation of the prefilter gain Ssr in phase variable canonical form. from a given transfer function (SISQ) Polar from Cartesian coordinates Pole assignment for a single input system by use of a Hessenberg algorithm Uniformly distributed random numbers Solution of a system of ftrst order di1ferential equations Solution of a system of ftrst order differential equations Solution of a system of ftrst order di1ferential equations. :fixed step integration Pole assignment for a single input system Sum of matrices QR factorization of a rectangular matrix Sorts a vector in increasing order Global minimum of a multi variable function. gradient not required Local minimum along a given direction

DIMAS

name BDIAG BPADE

DAD

DCDIV DECHES ELTR ELTRN EMULSH EXCHNG

EXCQZS

EXTl

EXTC EXTD GSCHV

HlSLV

H3SLV

HQR1

HQR4

HSHMLT HSLV LUSLV LYAC

LYAD

NTNC NTND ORTR PADE

PADES

PERMUT QRSTEP QZHESM

section 2.1.3.02.c 2.1.5.21.

2.1.1.21.b

2.0.1. 2.1.4.13. 2.1.4.02.c 2.1.4.03. 2.1.1.21.b 2.1.3.41.

2.1.3.42.

S.1.2.11.c

S.1.2.11.a S.1.2.11.b S.1.2.11.c

2.1.2.13.

2.1.2.13.

2.1.3.02.c

2.1.3.02.c

2.1.4.02.c 2.1.2.13. 2.1.2.51. S.1.3.a

S.1.3.b

S.1.2.11.& S.1.2.11.b 2.1.4.03. 2.1.5.21.

2.1.5.21.

2.0.3. 2.1.3.02.b 2.1.3.02.f

-62 -

description Transformation of a real Schur form to block-diagonal form Computes the exponential of & matrix by block diagonalization and rational Pade approximations Products aAD or aDA with a a real scalar. A arbitrary and D a matrix with ones down the minor diagonal Complex division in real arithmetic LR decomposition of & Hessenberg matrix Applies the transformation of ELMHES (EISPACK) Accumulation of similarity transformations Product of two real Schur matrices Reordering of Schur form for invariant subspace with prescribed spectrum Reordering of generalized Schur form for deftating subspace with prescribed spectrum Constructs the matrices de1ining the generalized eigenvalue problem for solving the (near) singular CAREIDARE Constructs the Hamiltonian matrix for solving CARE Constructs the extended symplectic matrix for solving DARE Solves CAREIDARE by the generalized Schur method (Van Dooren) Linear equations solution. matrix with two nontrivial lower subdiagonals Linear equations solution. matrix with three nontrivial lower subdiagonals Real Schur decomposition of a real upper Hessenberg matrix and accumulation of the similarity transformations performed Real Schur decomposition of a real upper Hessenberg matrix and accumulation of the orthogonal transformations performed Applies the transformation of ORTHES (EISPACK) Linear equations solution. Hessenberg matrix Solution of an inhomogeneous matrix equation Solution of ATX + XA - C. A upper Schur form. C symmetric Solution of AT XA + C - X. C symmetric. A upper Schur form Solution of CARE (iterative Newton) Solution of DARE (iterative Newton) Accumulation of orthogonal transformations Computes the exponential of a matrix by rational Pade approximations Computes the exponential of a real Schur matrix by rational Pade approximations Rearranges a vector with a given permutation One implicit QR step on an upper Hessenberg matrix Transformation of a pair of arbitrary matrices to upper Hessenberg and triangular form respectively

name QZITM

QZVAlM

QZVECM

SCHV SEIG SEORl SEOR2 SOLHES

SPLIT

SVEC SYLHC SYLHD SYLSC

SYLSD

SYLSM

UTAU

section 2.1.3.02.f

2.1.3.02.f

2.1.3.32.

5.1.2.11.c 2.1.3.15. 2.1.3.02.a 2.1.3.02.a 2.1.2.51.

2.1.3.02.c

2.1.3.15. 5.1.4. 5.1.4. 5.1.4.

5.1.4.

5.1.4.

2.1.1.21.b

- 63-

description Transformation of an upper Hessenberg matrix and a triangular matrix to a quasi-triangular and a triangular matrix respectively Transformation of an upper Hessenberg matrix and a triangular matrix to a quasi-triangular and a triangular matrix respectively Determines some eigenvectors of the generali2ed eigenproblem with A in upper Schur form and B in upper triangular form Solves CAREIDARE by the Schur vector method (Laub) Computes the eigenvalues of a Schur matrix Orders the eigenvalues of a quasi-triangular matrix Orders the eigenvalues of a quasi-triangular matrix Solution of an inhomogeneous matrix equation with a Hessenberg matrix. decomposed by DECHES Splits a 2 X 2 diagonal block of an upper quasi-triangular matrix Computes the eigenvectors of an upper Schur matrix Solution of AX + XB = C, (Hessenberg Schur method) Solution ofAXB + X-C. (Hessenberg Schur method) Solution of AX + XB ... C (Hessenberg Schur method). A lower. B upper quasi-triangular matrix Solution ofAXB + X ... C (Hessenberg Schur method), A lower. B upper quasi-triangular matrix Solution of AX + XB - C (Hessenberg Schur method), A lower. B upper Schur form. with exit if the magnitude of any solution element exceeds a given bound Product of UT AU, A symmetric and U upper triangular

BIMASC

name BALRNM DFLOAT DQRSLT

DUALS

EXTI

GAIN GAIN 1 GSTEP

H12 MREDUC

MTFF

MTVAR

MULVA MZEROS OPTR

ORTEQ

OUTP

REMC1

REMIN

REMOl RENEM

RENEMC

RENEMO

RKF45

RPCAR RPC02 RPDIV RPMUL RPOMD RPVAR SAESTM

SALOC

section 3.9. 2.0.1. 2.3.2.

3.1.21.

3.1.51.

4.1.8.21. 4.1.8.21. 4.1.15.

2.1.4.02.a 4.1.8.11.

5.1.14.

4.1.8.21.

2.1.1.31.a 4.1.8.11. 5.1.10.

4.1.2.

4.1.15.

4.1.1.

3.1.31.

4.1.1. 3.1.41.

3.1.41.

3.1.41.

2.5.1.21.

2.1.3.03. 2.2.1.11. 2.2.1.22. 2.2.1.22. 2.2.1.22. 2.2.1.31. 5.1.7.

5.1.1.a

- 64-

description Balances the subsystem of a time-invariant ss-model Conversion from integer to double precision Least squares solution of an over- or under-determined linear system Finds the dual (transpose) system of a given linear timeinvariant ss-model Connection of an internal model to the output of an openloop state space system Gain (SISO) of a transfer function from its ssr Gain (SIS0) of a transfer function from its ssr Evaluates the right-hand side of the linear system of differential or difference equations corresponding to various multivariable control structures Constructs and applies a Householder transformation Determines a reduced system of a ssr having the same system zeros Computes the disturbance and reference feed-forward gain matrices for a time-invariant system described by a transfer function matrix Computes the steady state gains (MIMO) for a given transfer function matrix Matrix times vector plus vector Computes the invariant zeros of ass-model Optimal state feedback gain matrix from the solution of a CAREIDARE Applies a general orthogonal system similarity transformation to a state space description Evaluates y - C xx + D xu. where y. x and u are the system output. state and input vectors. respectively Constructs a minimal order SSt in controllable canonical form Determines a reduced order ss-model from a non-minimal ss-model Constructs a minimal order ssr in observable canonical form Determines a non-minimal. uncontrollable and unobservable ssr for a given transfer matrix Determines a non-minimal. controllable ssr for a given transfer matrix Determines a non-minimal. observable ssr for a given transfer matrix Solution of a nonstiff or mildly stiff system of first order differential equations Characteristic polynomial of a Hessenberg matrix Computes the coefficients of a polynomial from the roots Quotient of two polynomials Product of two polynomials GCD of two polynomials Value of a polynomial in a given point Minimal order state estimator for a time-invariant continuous or discrete system Pole assignment by state feedback using the Schur method

name SIMEQ

SQRQDC STAC STAD STATE

STFF

TMICOl

TMTCD

TSBAL

TSBALI TSCD

TSCO TSMT

TSMTl

section 4.1.2.

2.1.4.21. 5.1.1.b 5.1.1.b 4.1.15.

5.1.14.

4.1.1.

3.5.

3.9.

3.9. 4.1.4.

3.1.61. 3.1.12.

3.1.12.

- 65-

description Applies a general system similarity transformation to a state space description RQ decomposition of a square matrix Computes a stabilizing gain matrix. continuous system Computes a stabilizing gain matrix. discrete system Evaluates Axx + Bxu + BZxw, where x. u and ware the system state, control and disturbance vectors, respectively. and A is a square matrix in upper Hessenberg form Computes the disturbance and reference feed-forward gain matrices for a time-invariant system described by a 88-

model Standard controllability form of a single input linear 88-

system by stabilized elementary similarity transformations Sampled-data system corresponding to a continuous system given by a transfer matrix Reduces a given ss-representation to numerically balanced form Computes the balancing transformation Computes the sampled data system corresponding to a given continuous ssr Reduces a matrix pair to the standard controllability form Finds the transfer function matrix of a given ssr. using orthogonal transformations Finds the transfer function matrix of a given ssr. using stabilized elementary similarity transformations

BYERS

name BCKSLV

DEPSLN ERF FNRMl FROB G3REF GENREF GRAND HAMHES HAMIT HAMQR HAMSCH HQRIT

LOWHES

LSCHUR LYAPUN MULTSF NEWTON ORDER PERMUT PRMBAK

QLIT

QLORDR QLSTEP RESID ROBERT RSCHUR S3REF

SCHUR2 SEPEST SNRMl SQRED SYFROB SYMEQU SYMREF UNBAL URAND V3REF VECREF

section 5.1.3.a

2.0.1. 2.0.1. 2.1.1.21.c 2.1.1.21.c 2.1.4.02.a 2.1.4.02.a 2.0.4. 2.1.3.51. 2.1.3.51. 2.1.3.51. 2.1.3.51. 2.1.3.02.c

2.1.3.02.c

2.1.3.02.c 5.1.3.a 2.1.1.22.b 5.1.2.11.a 2.1.3.51. 2.1.3.02.a 2.1.3.21.

2.1.3.02.c

2.1.3.02.a 2.1.3.02.b 5.1.2.11.a 5.1.2.11.a 2.1.3.02.c 2.1.4.02.a

2.1.3.02.c 2.1.3.61. 2.1.1.21.c 2.1.3.51. 2. 1. 1.21.c 2.1.4.42. 2.1.4.02.a 2.1.3.02.a 2.0.4. 2.1.4.02.a 2.1.4.02.a

- 66-

description Solution of ATX + XA == C. A quasi-triangular. C symmetric Relative unit roundoff quantity Errorfunction Ll-norm of a square matrix Frobenius norm of a square matrix Constructs a reflection of length 2 or 3 Constructs a reflection Normal (0.1) random number generator Reduction to Hamiltonian - Hessenberg form Hamiltonian QR iteration Hamiltonian QR step Hamiltonian-Schur decomposition Real Schur decomposition of a real upper Hessenberg matrix (mod. of HQR2. EISPACK) Transformation of an arbitrary (sub)matrix to lower Hessenberg form Transformation of a lower triangular Schur decomposition Solution of ATX + XA ... C. A arbitrary. C symmetric Product of a symmetric and arbitrary matrix Solution of CARE (Newton) Orders the eigenvalues of a Hamiltonian triangular matrix Isolates eigenvalues (mod. of BALANC. EISPACK) Backtransformation of Schur vectors from permuted (PERMUT-BYERS. see 02a) to original matrix Transformation of a lower Hessenberg matrix to lower quasi-triangular matrix Orders the eigenvalues of a quasi-triangular matrix A single QL step Residual of an approximate Riccati solution Solution of CARE with matrix sign function Real Schur decomposition (mod. of RG. EISPACK) Symmetric similarity transformation by a reflection of length 2 or 3 Schur decomposition of a 2 X 2 matrix Estimates sep(TT. -T) Ll-norm of a symmetric matrix Hamiltonian matrix to square reduced Hamiltonian matrix Frobenius norm of a symmetric matrix Equivalence transformation of a symmetric matrix Symmetric similarity transformation by a reflection Decodes and applies the transformation of BALANC Uniform random number generator Applies a reflection of length 2 or 3 to a set of vectors Applies a reflection to a set of vectors

DSP

name AMODSQ AUTO CCEPS CHEBC CHEBY CLHARM

CMFFr COVAR COVAR1 COVARl

COVLAT CZT DIFILT DODIF EXCH FAST FFA FFS

FFr

FFT2I FFTlT FFr842 FFrAOH

FFrASM FFTMX FFTOHM FFTSOH FFTSYM FOUREA FSST

HAMMIN ICCEPS IDEFIR

IFTAOH

IFTASM IFTOHM IFTSOH IFTSYM

KAISER LPTRN

LREMV

section 2.0.1. 6.8.11. 6.5. 6.9. 6.9. 6.8.22.

6.6.21. 6.8.21. 6.8.01. 6.8.01.

6.8.22. 6.7. 6.1. 6.1. 2.1.1.11.d 6.6.11. 6.6.11. 6.6.11.

6.6.12.

6.6.21. 6.6.21. 6.6.11. 6.6.11.

6.6.11. 6.6.11. 6.6.11. 6.6.11. 6.6.11. 6.6.11. 6.6.11.

6.9. 6.5. 6.10.a

6.6.11.

6.6.11. 6.6.11. 6.6.11. 6.6.11.

6.9. 6.8.01.

6.3.

- 67-

description Square of the modulus of a complex number Linear prediction analysis using the autocorrelation method Complex cepstrum of a sequence Chebyshev window parameters Dolph Chebyshev window design Linear prediction by a covariance lattice routine for harmonic mean method Optimized mass storage complex FFr Linear prediction analysis using the covariance method Covariance matrix of a given signal M+ 1) x (M+ 1) covariance matrix using the M X M covariance matrix and the signal General covariance lattice algorithm for linear prediction CHIRP Z-transform Decimation. interpolation or filtering of a signal Optimal digital interpolating filter Swaps vectors Finite discrete Fourier transform (DFT) of a real vector Finite DFT for a real vector (radix 8 algorithm) Fourier synthesis for a real vector from the Fourier coefficients (radix 8 algorithm) Multivariate complex Fourier transform. using mixed radix algorithm Two-dim. IFFT for real/complex data Two-dim. FFr for real/complex data Finite DFT for complex data DFT for a real. anti-symmetric. odd harmonic. N-point sequence DFT for a real. anti-symmetric. N-point sequence Mixed radix fast Fourier transform DFT for a real. odd harmonic. N-point sequence DFT for a real. symmetric, odd harmonic. N-point sequence DFT for a real. symmetric. N-point sequence Cooley-Tukey fast Fourier transform Fourier synthesis of a real vector from the Fourier coefficients Generalized Hamming window The inverse complex cepstrum Design of linear phase FIR-filters in direct form with minimum coefficient word length IDFT for a real, anti-symmetric. odd harmonic. N-point sequence IDFf for a real. anti-symmetric. N-point sequence IDFf for a real. odd harmonic. N-point sequence IDFf for a real. symmetric. odd harmonic. N-point sequence Inverse discrete Fourier transform (IDFT) for a real. symmetric. N-point sequence Kaiser window Transformations between various parameter sets used in linear prediction Removes the DC component and slope of a signal

name MXFLAT

NORMAL R1UNIF RADIX4

RANBYT RCEPS REALS

REALT

REMEZ

RFILT

RMFFT

SET SMINVD SORTG SRCONV

TRIANG UN! WFfA XFR ZERO

section 6.10.a

2.0.4. 2.0.4. 6.6.11.

2.0.4. 6.5. 6.6.12.

6.6.12.

6.10.a

6.10.a

6.6.21.

2.1.1.11.d 2.1.2.31.b 2.0.3. 6.1.

6.9. 2.0.4. 6.6.11. 2.1.1.11.d 2.1.1.11.e

- 68-

description Coefficients of a maximally flat FIR linear phase filter with odd number of terms and even symmetry in filter coefficients Generates an independent pair of random normal deviates Uniform random number generator both in real and in bits Time-efficient forward or inverse complex DFf via radix 4 FFf Uniform random number generator both in real and in bits Real cepstrum of a real sequence With FFf to compute Fourier transform or inverse for real data With FFf to compute Fourier transform or inverse for real data. single- or multivariate Remez exchange algorithm for the weighted Chebyshev approximation of a continuous function with a sum of cosines Filters one frame of data for a given filter frequency response Optimized multidimensioned mass storage FFT. real to complex or vice-versa Makes a copy of a vector Inverse of a positive definite matrix Sorts a vector in increasing order Conversion of the sampling rate of a signal by the ratio LIM Triangular window Uniform random number generator Winograd Fourier transform Makes a copy of a vector Sets each element of a vector to zero

KONTOS

name DIVIDB

INVUM RCONS RDIV REDUCE RMUL SPDEC

SPFE UM ZPOLM

- 69-

I section I description 2.2.3. Solves A(X) .. F(X)B(X). with polynomial vector A(X) and

2.2.3. 2.2.3. 4.2.3. 2.2.3. 2.2.3. 2.2.3.

4.2.8. 2.2.3. 4.2.8.

polynomial BCX) of at most second degree. given Inverse of a polynomial matrix Product of elementary factors Greatest right divisor of a rectangular pm Computes an elementary factor of a pm with a given spectrum Product of two polynomial matrices Determines P and Q such that det(A(X» ... det(XP - Q) for a given A(X) Right divisor of a pm with zeros in a given region Augments a full row rank. pm to a unimodular pm All zeros of a polynomial matrix

LISPACK

name CONDIT EXCHQR

EXCHQZ

HQRNOZ LYCSL LYCSR LYDSL

LYDSR

ORDERS

ORDERZ

PADE POLSC QRSTEP

QZSTEP

RICSL

RIDSL

SOLCS

SOLDS

SYCSL

SYCSR

TRMCF

TRSCF

section 2.1.3.61. 2.1.3.41.

2.1.3.42.

2.1.3.61. 5.1.3.a 5.1.3.a 5.1.3.b

5.1.3.b

2.1.3.41.

2.1.3.42.

2.1.5.21. 5.1.1.a 2.1.3.41.

2.1.3.42.

5.1.2.11.a

5.1.2.11.b

5.1.2.11.a

5.1.2.11.b

5.1.4.

5.1.4.

4.1.1.

4.1.1.

- 70-

description Computes the condition number of an eigenvalue Reordering of Schur form for invariant subspace with prescribed spectrum Reordering of generalized Schur form for de1lating subspace with prescribed spectrum Computes the condition number of an eigenvalue Solution of ATX + XA == C, A arbitrary, C symmetric Solution of ATX + XA "" C, A arbitrary, C symmetric Solution of ATXA - X '" C, C symmetric (Barraud's method) Solution of ATXA - X == C, C symmetric (Barraud's method) Reordering of Schur form for invariant subspace with prescribed spectrum Reordering of generalized Schur form for deflating subspace with prescribed spectrum Computes the matrix exponential with accuracy estimate Pole assignment synthesis Reordering of Schur form for invariant subspace with prescribed spectrum Reordering of generalized Schur form for deflating subspace with prescribed spectrum Optimal control via the matrix Riccati equation (continuous systems) Optimal control via the matrix Riccati equation (discrete systems) Optimal control via the matrix Riccati equation (continuous systems) Optimal control via the matrix Riccati equation (discrete systems) Solution of AX + XB = C (Bartels-Stewart method). nothing known about A. B. C Solution of AX + XB .. C (Bartels-Stewart method). nothing known about A, B. C Reduction of a multi input system into canonical form by orthogonal transformation Reduction of a single input system into canonical form by orthogonal transformation

LPS

name AUTO 1

AUT02

COVARl

COVAR2

DIRECT EVAL

FFT KLOCH ONEMUL STEPDN STEPUP TWOMUL

section 8.2.

8.2.

8.2.

8.2.

8.2. 8.2.

6.6.11. 8.2. 8.2. 8.2. 8.2. 8.2.

-71-

description Linear predictor polynomials and reflection coefficients by autocorrelation method Linear predictor polynomials and reflection coefficients by autocorrelation method Linear predictor polynomials and reflection coefficients by covariance method Linear predictor polynomials and reflection coefficients by covariance method Output of a lattice filter applied to a time series Transformation of a rational function to reflection coefficients and tap weights Radix 2 fast Fourier transform Output of a lattice filter applied to a time series Output of a lattice filter applied to a time series Reflection coefficients from predictor polynomials Predictor polynomials from reflection coefficients Output of a lattice filter applied to a time series

RASP

name ABTAST ADD AESTE AMTM AMTM ARRAY

ARRAYS

ATAXY ATAXYD ATXPXA

AUD BOLAUB BOPLOT BOPLOT COMPOR CONOBS CSTAB DEFIT DFP

DIADD DIWI DOMWES DPND DRICLB

DRICNT

DRINWT DSTAB EATl EAT4 EIGEN

EITEST ELDIN ENTKOP EQUATE EQUIL FACfOR

FACfOR FEHDIM FMIN

FRELOG

FROB

section 4.1.4. 2.1.1.22.a 4.3.8. 2.1.1.21.b 2. 1. 1.22.b 2.1.1.01.

2.1.1.01.

2.0.1. 2.0.1. 5.1.3.a

5.1.2.11.b 4.1.14. 4.1.14. 4.2.12. 2.0.3. 4.1.6. 5.1.1.b 2.1.2.01. 2.3.5.12.

2.1.1.22.a 5.1.10. 4.1.9. 2.1.1.01. 5.1.2.11.b

5.1.2.11.b

5.1.2.11.b 5.1.1.b 2.1.5.11. 2.1.5.11. 2.1.3.11.a

2.1.3.16. 4.1.1. 4.1.3. 2. 1. 1.21.d 3.9. 2.1.2.11.b

2.1A.11.b 2.1.1.01. 2.3.5.12.

4.2.12.

4.1.1.

-72-

description Discrete time from continuous time ss-model Sum of matrices Number of cuts of a root locus with a rectangular frame Product AB. ABT. ATB. ATbT from A Band submatrices Product of matrices Matrix storage from one- to two-dimensional array. vice versa Matrix storage from one- to two-dimensional array. vice versa Angle in polar from Cartesian coordinates Angle in polar from Cartesian coordinates Solution of ATX + XA ... C. A arbitrary or Schur form. C symmetric Hamiltonian matrix for solving the Riccati equation Logarithmic frequency response of ass-model Bode diagrams Plots a Bode diagram Sorts a one-dimensional array of complex numbers Controllability and observability test of ass-model Computes a stabilizing gain matrix. continuous system Test on definiteness of a matrix Global minimum of a multivariable function. gradient required Matrix plus constant times unity matrix Weighting matrices for the linear optimal sampled data Dominance measures Linear dependency test of vectors Solution of DARE (Laub) and optimal steady state feedback gain Solution of DARE (Kleinman) with stability margin assignment Solution of DARE (Kleinman) Computes a stabilizing gain matrix. discrete system Computes the matrix exponential and its integral Matrix exponential and integrals of it Computes all eigenvalues and eigenvectors (if desired) of an arbitrary matrix by routines of EISPACK Accuracy test of eigenvalues and eigenvectors Computes the FIN-form of a given SSt

Computes the decouplingsindices of a system in FIN-form Makes a copy of a matrix Balances a linear ss-model Linear equations solution, positive definite matrix. decomp. given . Cholesky decomposition of a positive definite matrix Compatibility test of two matrices Global minimum of a multivariable function. gradient required Gain and phase in a given interval (log) from a scalar factorized transfer function. discrete or continuous Computes the FN-form of a ssr given in FIN-form

name G1 G2 GAUSS GDIVS GESYSO

H12 HESPOL HOMGA2 HOUS HQR2

HQR3

IKL INDEX INSEDS INSERT INTEAT

INV ITERR JUXTC JUXTR KONFIG KPOL KRICLB

KRICNT

KRINWT KRIPLO LDLT LDP LDPEI LOESHO LOESIN LSQ MAKODD MAMUDD MAMUDD MARKOV MASEDD MASEDS MAXEL MAXIND MINF MNUL2D MODOM

MPGRD MSCALE

section 2.1.4.02.b 2.1.4.02.b 2.0.4. 2.0.1. 5.5.

2.1.4.02.a 2.1.3.03. 5.5. 4.1.0. 2.1.3.14.

2.1.3.02.c

2.0.1. 4.1.3. 2.1.1.21.d 2.1.1.21.d 4.1.15.

2.1.2.31.a 2.1.1.21.c 2.1.1.21.d 2.1.1.21.d 4.3.8. 5.1.1.a 5.1.2.11.a

5.1.2.11.a

5.1.2.11.a 4.3.8. 2.1.4.01. 2.3.8.21. 2.3.8.21. 2.1.2.51. 2.1.2.51. 2.3.6. 2. 1. 1.21.d 2.1.1.21.b 2.1.1.22.b 3.8.b 2.1.1.21.d 2.1.1.21.d 2.1.1.21.c 2.1.1.11.c 2.3.1.21. 2.1.1.22.e 4.1.9.

2.2.3. 2.1.1.21.a

- 73-

description Constructs a Givens plane rotation Applies a Givens plane rotation Gaussian distributed random numbers Divisibility test of two real numbers Connects control loop and dynamic regulator into a total system Constructs and applies a Householder transformation Characteristic polynomial of a Hessenberg matrix System-expansion with a homogeneous inputsignal generator Elementary orthogonal transformation of a linear system All eigenvalues and eigenvectors of an upper Hessenberg matrix (mod. of HQR2. EISPACK) Transformation of a Hessenberg form to Schur form with ordered eigenvalues Entier of a real number Controllability and Kronecker indices Composition of blockmatrices Composition of blockmatrices Solution of a continuous time invariant linear system with transition matrix Inverse and determinant of a full arbitrary matrix Measure of the difference of two matrices Composition of matrices. column-wise Composition of matrices. row-wise Configuration of poles and zeros for root loci Feedback gain for eigenvalue assignment Solution of the continuous algebraic matrix Riccati equation (CARE) (Laub). and optimal steady state feedback gain Solution of CARE (Kleinman) with stabilization according to Armstrong Solution of CARE Plots the pole-zero configuration in a root loci diagram Update of a Cholesky decomposition Linearly constrained problems. quadratic programming Linearly constrained problems. quadratic programming Solution of a homogeneous matrix equation Solution of an inhomogeneous matrix equation Solution of a linear least squares problem Modification of a (sub)matrix Product AB. ABT. ATB. ATbT from A Band submatrices Product of matrices Markov parameters of a given ssr Composition of matrices Composition of matrices. double to single precision Maximum element of a matrix Maximum element of a vector Onedimensional minimum along a given direction Initialization of null-matrix Dominance analysis of the eigenvalues of a modal transformed system Degree of a matrix polynomial Scalar times matrix

name MULT MULT NNLS NORMSl NORMSS NULL NYPLOT

NYPLOT

ORDNE2 ORTFUN

PART PEXMA PMSAD PMULT POLCOF POLCOS POLDIV POLMUL POLSAD POLVAL POSIHE POWELL

PREFIL

PTRA

PUNKTE QMQP QUAD RATION REDHN REQUIL RICAT RUND SAMDA

SAMPL

SBEIND SLLSPQ

SMITH SNVDEC

SORT SPRANT SUBT SUM

section 2.1.1.21.b 2.1.1.22.b 2.3.6. 2.1.1.22.c 2.1.1.22.c 4.1.0. 4.3.9.

4.2.11.

2.0.3. 4.3.9.

2.1.1.21.d 2.2.3. 2.2.3. 2.2.3. 2.2.1.11. 2.2.1.11. 2.2.1.22. 2.2.1.22. 2.2.1.21. 2.2.1.31. 5.1.1.a 2.3.5.12.

5.1.10.

4.1.0.

4.3.9. 2.2.3. 2.4.1. 2.2.2. 4.1.6. 3.9. 5.1.2.11.a 2.0.1. 5.1.10.

8.1.a

4.1.3. 2.3.10.

5.1.4. 2.1.4.31.

2.0.3. 4.1.15. 2.1.1.22.a 5.1.3.b

- 74-

description Product AB, ABT, ATB. ATbT from A Band submatrices Product of matrices Solution of a nonnegative linear least squares problem Norms of matrices. Lp. p==1, 2. co. Frobenius Norms of matrices. Lp. p==1. 2. co. Frobenius Annulates small elements in the matrices of ass-model Draws Nyquist and Popov plots and frequency loci for ztransformations Nyquist diagrams of scalar transfer functions of discrete or continuous systems Sorts a one-dimensional array Computes complete Nyquist. Popov. Tsypkin plots for a rational transfer function Partitioning of a matrix Degree of an element of a polynomial matrix Sum or difference of two polynomial matrices Product of two polynomial matrices Computes the coefficients of a polynomial from the roots Computes the coefficients of a polynomial from the rootS Quotient of two polynomials Product of two polynomials Sum or difference of two polynomials Value of a polynomial in a given point Feedback vector for pole placement Global minimum of a multivariable function. gradient not required Elimination of cross-product term in the quadratic performance index Elementary transformation with permutation of a linear system Computes a complete Nyquist plot Conversion of a matrix polynomial into a polynomial matrix Computes the zeros of a quadratic function Value of a rational function in a given point Controllable or observable part via HN-form Backtransformation of a balanced system Solution of CARE with stability margin assignment Rounding a number to absolute or relative precision Linear optimal sampled data regulator from linear optimal continuous regulator Solution of a matrix Riccati difference equation for discrete Kalman filter and Kalman gain Controllability. observability or decouplingsindices Sequential linear least squares programming to solve a general nonlinear optimization problem Solution of AP + PB - - Q. A and B arbitrary but stable Singular value decomposition of an arbitrary rectangular matrix Sorts a one-dimensional array of integers Step response of a continuous or discrete linear system Difference of matrices Solution of AXe + B - X

name SYMPDS SYMPDS SYMSLV SYSAT SYSRED

TFEIG TFFAHN

TFLAUB TFMRP TFPART TRANP TRCE UMORD UNITY URAN WEZU WIFUl WIFUD WOKl WOKPUl WOPLOT ZPFORD ZPFORM ZRPOLY ZUSTD

section 2.1.2.51. 2.1.4.11.b S.1.3.a 4.1.15. 4.1.9.

3.1.31. 3.3.

3.6. 4.3.6. 4.1.8.11. 2.1.1.2l.b 2.l.l.22.f 4.1.0. 2.1.1.21.e 2.0.4. 4.1.9. 4.3.8. 4.3.8. 4.3.8. 4.3.8. 4.3.8. 2.0.1. 2.0.1. 2.4.1. 3.1.41.

- 75-

description Solution of a matrix equation with a positive definite matrix Cholesky decomposition of a positive definite matrix Solution of AT X + XA ... C, A quasi-triangular. C symmetric System response of a continuous linear system Model reduction of a modal transformed system by dominant mode analysis Transfer matrix from poles and residues Finds the dual right(left) polynomial matrix representation (pmr) Real and imaginary part of a matrix frequency response Poles and zeros of a transfer matrix Poles and residues of ass-model Transpose of a matrix Trace of a matrix Permutation of states Initialization of a matrix by a unity matrix Uniform random number generator Evaluation of dominant states and eigenvalues Angular function evaluation for root loci Angular function evaluation for root loci Root loci curves Root loci points Drawing of root loci Mantissa and exponent of a number in double precision Mantissa and exponent of a real number Computes the zeros of a real polynomial Rational transfer function to system matrix form

SliCE

name BALRS

CHOLD DECZR HHDME

HHDML

LYBAD

LYBSC

MATM MCINX

MEEIG

MEINT MEPAD

MRINX

PMXDL

PMXFR

PMXSS POLSC

QTRORT

RILAC SPDLY

SSBAL

SSCASC

SSFEED

SSPARA

SSTZER SSXDL

SSXFR SSXKF

section 2.1.3.02.a

2. 1.4. 11.c 3.1.61. 2.1.4.41.

2.1.4.41.

5.1.3.b

5.1.3.a

2.1.1.21.b 2.1.3.33.

2.1.5.11.

2.1.5.1l. 2.1.5.21.

2.1.3.33.

3.3.

3.3.

3.3. 5.1. La

2.1.3.02.c

5.1.2.11.a 5.1.3.a

3.9.

3.1.5l.

3.1.51.

3.1.51.

4.1.8.11. 3.1.21.

3.1.11. 2.1.3.33.

- 76-

description Balances an arbitrary matrix in order to minimize its maximum norm Symmetric matrix. Cholesky decomposition. semi definite. Reduction to staircase form with triangular pivots Pre- or post multiplication of an arbitrary matrix by an orthogonal matrix . Pre- or post multiplication of an arbitrary matrix by an orthogonal matrix Solution of ATXA - X-C. C symmetric (Barraud's method) Solution of AT X + XA - C. A arbitrary or Schur form. C symmetric (Bartels-Stewart method) Product of matrices AB. ABT. ATB. ATbT

Computes the Kronecker column indices and the infinite elementary divisors of an M by N pencil AB - A Computes the matrix exponential of a real non-defective matrix with real or complex eigenvalues Computes the matrix exponential and its integral Computes the exponential of an arbtrary matrix using a Pade approximation Computes the Kronecker row indices and the infinite elementary divisors of an M by N pencil A B - A Finds the dual right(1eft) polynomial matrix representation (pmr) Computes the transfer matrix of a left or right pmr at a given frequency Finds a ssr equivalent to a given left or right pmr Determines the state feedback matrix of a linear timeinvariant single-input system in ssr such that the closedloop system has desired poles Transformation of an arbitrary (sub)matrix or upper Hessenberg form to quasi-triangular form Solution of CARE (Laub's Schur form method) Solution of ATX + XA .... C. A abitrary. C factorized as BTB Reduces a given ss-representation to numerically balanced form Computes the ssr for the cascaded interconnection of two ss-systems Computes the ssr for the feedback interconnection of two ss-systems Computes the ssr for the parallel interconnection of two ss-systems Computes the invariant zeros of ass-model Finds the dual (transpose) system of a given linear timeinvariant ss-model Computes the complex frequency response matrix Computes Kronecker indices and all elementary divisors of an M by N pencil A B - A

name SSXMC

SSXMR SSXPM

SSXSC

SSXTM SSZER SYHSC TFXFR

TMXPM

TMXSS

section 3.1.11.

3.1.13. 3.1.13.

3.1.11.

3.1.12. 4.1.8.11. 5.1.4. 3.5.

3.5.

3.5.

- 77-

description Reduces a time-invariant multi-input system to orthogonal canonical form Finds a minimal ssr (staircase form) for a given ssr Finds a relatively prime left or right pmr which is equivalent to a given ssr Reduces a time-invariant single-input system to orthogonal canonical form Finds the transfer function matrix of a given ssr Computes the invariant zeros of ass-model Solution of AX + XB - C. (Hessenberg Schur method) Computes the value of a complex valued rational transfer function for a given frequency (SIS0) Finds a relatively prime left or right pmr for a given proper transfer function matrix (MIMO) Finds a minimal ssr for a given proper transfer function matrix

SYCOT

name ACCOR APMB ARMAH ASOLVE ASVD

ATA ATBA ATSA ATXPXA

AXB BCKMLT BILNTR BLNC BOX

CGENMR

CHANKX

CLS CLYA CONTRL CROUT

CSRF DCHESS DCNORM DDEADB DDEADB DDSUBS

DEXCHQ

DFASI DGIV DLYA DMAX DMIN DNREF DNREFG DNRM2 DPLMMA DPYTAG DSREF DSREFG DSSQ DSTAIR DSTAIR DSTAIR

section 6.4. 2.1.1.22.a 3.8.b 2.3.2. 2.1.4.31.

2.1.1.22.b 2.1.1.21.b 2.1.1.21.b 5.1.3.a

2. 1. 1.22.b 2.1.3.21. 4.1.4. 3.9. 2.3.5.12.

3.8.b

3.8.a

7.2.1. 5.1.3.a 4.1.6. 2.1.2.11.a

8.1.b 3.1.11. 2.1.1.22.c 4.1.6. 5.1.6. 2.1.3.42.

2.1.3.42.

2.1.4.11.c 2.1.4.02.b 5.1.3.b 2.1.1.11.c 2.1.1.11.c 2.1.4.02.a 2.1.4.02.a 2.1.1.11.c 2.3.4. 2.0.1. 2.1.4.02.a 2.1.4.02.a 2.1.1.11.c 3.1.61. 4.1.1. 4.1.6.

- 78-

description Correlation coeffients between two multivariable sequences Sum or difference of arbitrary matrices Markov parameters of a multivariable ARMA-model Adaptive LS solution Singular value decomposition of a large matrix with low rank Product: matrix and its transpose Product of AT BA. A and B arbitrary Product of XT QX. X arbitrary. Q symmetric Solution of ATX + XA = C. A arbitrary or Schur form. C symmetric Product of matrices Transformation of eigenvectors Discrete time from continuous time ss-model. or vice versa Computes the balancing transformation of IABLNC Global minimum of a multivariable function. gradient not required Monovariable impulse response sequences H(n.a.cp •... ) for a pole of multiplicity n. damping a and sample angle cp Construction of the Hankelmatrix expansion of a multivariable parameter sequence Correlation least squares method Solution of AT X + XA ... C. A arbitrary. C symmetric Controllability test of a given ss-model Linear equations solution. arbitrary matrix. decomp. not given Square root Kalman filter (Chandrasekhar) Computes controller Hessenberg form Frobenius norm of the difference of two matrices Controllable subspace Dead beat control Reordering of generalized Schur form for deflating subspace with prescribed spectrum Reordering of generalized Schur form for deflating subspace with ,rescribed spectrum QTQ decomposition of a symmetric matrix Constructs a Givens plane rotation Solution of the discrete time Lyapunov equation Maximum element of a vector Minimum element of a vector Applies a Householder transformation Constructs a Householder transformation L2-norm of a vector Discrete piecewise linear minimax approximation Modulus of a complex number Applies a skew Householder reflection Constructs a skew Householder reflection L2-norm of a vector Reduction to staircase form with triangular pivots Transformation matrix for (upper)staircase form Reachable or unobservable subspace

name DTLLS DXTHAM EIGSYS EIGVA EIGWV ELS

EQROW EXCHNG

EXCHQR

FAC FADDEE

FCC FCD FFT FIR FMFP FNORM FSC FXFTPS

GCHOL

GCRD GDEADB GENMAR

GIV GLA

GLB

GLOVER

GPLIN HANKEX

HCORR

HDIFF HQRT

HQRT

HREAL

HSHLDR

IABLNC

section 2.3.2. 5.1.2.11.c 4.1.8.11. 4.1.8.11. 2.1.3.11.a 7.2.5.

2.1.1.11.e 2.1.3.41.

2.1.3.41.

6.4. 3.1.31.

6.4. 6.4. 6.6.11. 7.2.5. 2.3.5.12. 2.1.1.22.c 6.5. 5.1.3.b

2.1.4.11.c

4.2.3. 5.1.6. 3.8.b

2.1.4.02.b 7.2.5.

7.2.5.

4.1.9.

2.3.8.31. 3.8.a

3.8.b

5.1.9. 2.1.3.02.c

2.1.3.14.

5.1.9.

2.1.3.02.c

3.9.

- 79-

description Total LLS-solution ofAX=B. with A and B inaccurate Generalized Hamilton method (Van Dooren) Poles and zeros of ass-model Pole-zero map of a multivariable system All eigenvalues and eigenvectors of a major submatrix Extended least squares iteration. ARMA model for process. MA model for noise Sets each element of a vector to a constant value Reordering of Schur form for invariant subspace with· prescribed spectrum Reordering of Schur form for invariant subspace with prescribed spectrum Autocorrelation function of a signal Inverse of sl - A and the transfer matrix of a given ssmodel Crosscorrelation function of two signals Convolution and deconvolution product Mixed radix fast Fourier transform (complex signal) Least squares finite impulse response system identification Local minimum of a multivariable function Frobenius norm of an arbitrary matrix Sine and cosine transforms Solution of ATXA - X ... C. C symmetric (Barraud's method) Cholesky decomposition. semi definite. of a symmetric matrix Greatest common divisor of two pm's Generalized dead beat control Monovariable impulse response sequences H(n.a.¢ •... ) for a pole of multiplicity n. damping a and sample angle ¢ Constructs a Givens plane rotation Generalized least squares. low order noise. iterative technique. ARMA model for process. AR model for noise Generalized least squares. high order noise. iterative technique. ARMA model for process, AR model for noise Optimal Hankelnorm approximant of a balanced continuous ss-system Projected gradient method with upper and lower limits Construction of the Hankelmatrix expansion of a multivariable parameter sequence Impulse response from input/output data using the correlation method Accuracy check of the model obtained by realization Transformation of a Hessenberg form to Schur form with transformation matrix All eigenvalues and eigenvectors of an upper Hessenberg matrix Minimal state space model from a system given by its Markov sequence Transformation of an arbitrary (sub)matrix to upper Hessenberg form Balances (interactively) a not necessarily minimal ss-system

name INTGRA

INVERS INVMAT INVSUB

IOIR LDW

LINE LSA LSB LYAP MA11SM MARKOV MEAN MINMAX MLH

NOISE NORMM OBSER ORC ORDROW

OR! PAPT PRBS PSD QRSTEP RANK REC

RECUR

REDUCE RKF RLTR RPOLY RUN RUNDIS SHRSLV

SNREF SNREFG SORT SPLIT

SPYTAG SQAXB SQUAR1

section 2.5.1.21.

2.1.2.35. 2.1.2.31.a 2.1.3.41.

3.8.b 6.9.

2.3.1.21. 7.2.5. 7.2.5. 5.l.3.c 2.1.1.01. 3.8.b 2.1.1.11.c 2.1.1.l1.c 7.2.4.

2.0.4. 2.1.1.21.c 4.1.6. 7.2.10. 2.1.1.11.c

7.2.10. 2.1.1.22.b 2.0.4. 6.5. 2.1.3.02.b 2.1.2.01. 7.2.5.

7.2.5.

4.1.8.11. 8.1.a 6.6.11. 2.4.1. 4.1.4. 4.1.4. 5.1.4.

2.1.4.02.a 2.1.4.02.a 2.1.3.02.8 2.1.3.02.c

2.0.1. 2.1.1.21.b 8.1.b

- 80-

description Solution of a stiff system system of first order differential equations Generalized inverse of a general matrix Inverse of a matrix Reordering of Schur form for invariant subspace with prescribed spectrum Multivariable impulse response by deconvolution Data windowing of a correlation function: hanning window. hamming window. quadratic window Line search via parabolic interpolation Least squares estimation using pseudo inverse Least squares estimation using orthogonal functions Observability and controllability Gramians of ass-system Sum of the elements of an array Multivariable impulse response of a given ssr Mean of a vector Maximum-minimum element of a vector Maximum likelihood iteration of an ARMA model for the process and a MA model for the noise Generation of a noise sequence with given mean and variance Maximum element of a matrix Observability test of a given ss-model Product moment (determinant ratio) test Index of the largest vector component starting from a given index Instrumental Froduct moment test Prod uct: ASA • S symmetric Generation of a pseudo random binary noise sequence Powerdensity spectrum A single QR step Rank of a matrix Generalized least squares. recursive technique. ARMA model for process, AR model for noise Recursive techniques for ARMA process and MA noise: simple least squares. extended least squares. least squares with instrumental variable Computes the invariant zeros of ass-model Recursive Kalman filtering Mixed radix fast Fourier transform Computes the zeros of a real polynomial Solution of the continuous time ss-equations Solution of the discrete time ss-equations Solution of AX + XB ... C (Hessenberg Schur method). A lower. B upper Schur form Applies a Householder transformation Constructs a Householder transformation Orders the eigenvalues of a real Schur matrix Reduction of a 2 X 2 diagonal block of a real Schur matrix to upper triangular form Modulus of a complex number Product of matrices Kalman filter covariance. Hessenberg form

name SQUAR1

SQUAR2 SQUAR2

SRCF

SRIF SSFRBD

SSFRNY

SSOUT SSOUT2 SSSQ SSTRAN STLLS SVM

SWAPP

SYMSLV SYMSOL

TOEPEX

TRACE TRANSF TRANSP TRPS UNIMOD UUTR

XDRICC XRICCA ZERO

section 5.1.2.21.

8.1.b 5.1.2.21.

5.1.2.21.

8.1.b 4.3.10.

4.3.9.

3.7. 3.7. 2.1.1.11.c 4.1.4. 2.3.2. 7.2.4.

2.1.3.41.

S.1.3.a S.1.3.b

3.8.a

2.1.1.22.f 3.1.31. 2.1.1.22.b 2.1.1.21.b 2.2.3. 3.B.a

S.1.2.11.b S.1.2.11.a 4.1.8.11.

- 81-

description Finite interval discrete optimal control (dual to Kalmanfilter) Kalman filter covariance. Schur form Finite interval discrete optimal control (dual to Kalmanfilter) Finite interval discrete optimal control (dual to Kalmanfilter) Kalman filter information. Schur form Calculates a Bode diagram from a rational. continuous transfer function Calculates a Nyquist diagram from a rational. continuous transfer function Output sequence of a given SSt

Output sequence of a given SSt with a Hessenberg matrix L2-norm of a vector Discrete time from continuous time ss-model Total LLS-solution of AX-B. with A and B inaccurate Estimation of the continuous-time parameters of a state space model Reordering of Schur form for invariant subspace with prescribed spectrum Solution of ATX + XA "" C. A quasi-triangular. C symmetric Solution ofAXAT + C - X. C symmetric. A lower Schur form Computes a Toeplitz matrix expansion of a time sequence at a specified moment Trace of a matrix Rational transfer function of a SSt

Transpose of a matrix Transpose of a matrix Augments a full row rank pm to a unimodular pm Computes UUT • with U the Toeplitz matrix expansion of a given time sequence State space optimal regulator gain of DARE Solution of CARE (Laub's Schur form method) Computes the invariant zeros of ass-model

TIMSAC

name AICCOM

AMCOEF ARBAYS

ARCHEK ARMPIT

ARMLE

BAYSPC BAYSWT . BINARY CAN COR CANOCO

COMAIC COMPSD CONTRL COpy CPROCT FOUGER FUNCT FUNCT2

HESIAN

HUSHLl HUSHLD INVDET INVERS LINEAR LTINV MARFIT

MBYSAR MBYSPC MGSA MNONSB

MPARCO

MREDCT MSDCOM

MSETXl

MSVD

NONSTB NRASPE

section 7.2.0.a

7.2.0.b 7.2.3.

7.3.4. 7.2.4.

7.2.4.

7.2.3. 7.2.3. 2.0.!' 7.2.1. 7.2.1.

7.2.0.a 7.2.0.e 5.1.10. 2.1.1.11.d 7.2.0.d 6.6.11. 7.2.4. 6.2.

7.2.4.

2.1.4.21. 2.1.4.21. 2.1.2.31.a 2.1.2.31.e 2.3.1.21. 2.1.4.11.b 7.2.4.

7.2.3. 7.2.3. 2.1.4.21. 7.2.3.

7.2.5.

2.1.4.02.& 7.2.0.d

7.2.0.c

2.1.4.31.

7.2.3. 6.5.

- 82-

description Computes innovation variance and AIC of a model with M regressors Initial estimates of AR and MA coefficients Bayesian procedure with models of successively increasing order Checks the stability of the AR or MA part of a model Minimum AIC procedure with AR models of successively increasing order Exact maximum likelihood estimates of the parameters of an ARmodel Partial autocorrelation coefficients of the Bayesian model Bayesian weight of the AR model of each order Decimal to binary conversion Fits an ARMA model to stationary scalar time series Future canonical weights and the order of the Markovian model Innovation variance and AIC computation Residual variance of a subset regression model Optimal control from the gain matrices Makes a copy of a vector One step ahead prediction value of the controlled process Fourier transform (Goertzel method) Exact likelihood and its gradient of the m-th order AR model White noise variance and (-2)10g likelihood of a data sequence Inverse of an approximation to the Hessian of a loglikelihood function of the AR model of order k Performs the Householder transformation Transformation of a matrix into a triangular matrix Inverse and determinant of a full arbitrary matrix Inverse of a triangular matrix Linear search along a given direction Cholesky decomposition of a positive definite matrix Multivariate AR model fitting using the minimum AlC procedure Multivariate AR model fitting by a Bayesian procedure Partial AR coefficients of the multivariate AR model Modified Gram-Schmidt algorithm Multivariate AR model fitting to instationary time series by a Bayesian procedure Least squares estimates of partial AR coeffi.fient matrices of a multidimensional AR model Householder reduction One step ahead prediction error variance matrix for a multivariate AR model Prepares the data matrix for the fitting of a multivariate AR model Singular value decomposition of an arbitrary rectangular matrix Bayesian type non-stationary AR-model fitting procedure Power spectrum of an ARMA process

name OPTDES PERREG POLYRT PRDCfl PRDCf2

PRDCf3

PRDcr6

REDUcr RN RNOR SBBAYS SDCOMP SETLAG

SETXl SETX2 SETX4

SETX5 SETX6

SGRAD SMINOP SOLVE

SRCOEF

SRTMIN SUBPM

SUBSET SUBSPC

SYSTEM TRIINV WINDOW YMIN

section 5.1.10. 7.2.0.c 2.4.1. 7.2.0.d 7.2.0.d

7.2.0.d

7.2.0.d

2.1.4.02.a 2.0.4. 2.0.4. 7.2.3. 7.2.0.e 7.2.0.c

7.2.0.c 7.2.0.c 7.2.0.c

7.2.0.c 7.2.0.c

2.3.1.11. 7.2.4. 2.1.2.51.

7.2.0.e

2.0.3. 7.2.0.e

7.2.4. 7.2.3.

4.1.15. 2.1.2.31.e 6.9. 2.1.1.11.c

- 83-

description Optimal controller gain matrices Prepares the data matrix for the fitting of a periodic model Computes the zeros of a real polynomial by Newton-Raphson Several steps ahead prediction value of an ARMA model Several steps ahead prediction value of a non-linear regression model Several steps ahead prediction value of a non-linear regression model Several steps ahead prediction value of a non-linear regression model Householder reduction Uniform random number generator Normal (0.1) random number generator Bayesian model based on all subset regression models Residual variance of a regression model Prepares the specification of regressors for the fitting of (polynomial type) non-linear model Prepares the data matrix for AR model fitting Prepares the data matrix for non-linear AR model fitting Prepares the data matrix for the fitting of an AR model with polynomial type mean value Prepares the data matrix for a non-linear regression model Prepares the data matrix for a non-linear regression model with Laguerre type regressors Computes an approximation to the gradient by differentiation Controls the maximum likelihood computation Solution of an inhomogeneous matrix equation with an upper triangular matrix Subset regression coefficients and residual variance computation Sorts a vector in increasing order Variance matrix of a stationary state vector by the procedure of Akaike Minimum AIC type subset regression analysis Bayesian estimates of partial correlations by checking all subset regression model" Simulation of the control system Inverse of a triangular matrix Operates as a data window Minimum element of a vector

an inventory of basic software for computer aided control

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