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Copyright (C) 2004: R B Davies
28 March, 2004 - beta version.
| 1.
Introduction 2. Getting started 3. Reference manual |
4.
Error messages 5. Design of the library |
This is the how to use documentation for newmat plus some background information on its design.
There is additional support material on my web site.
Navigation: This page is arranged in sections, sub-sections and sub-sub-sections; four cross-references are given at the top of these. Next takes you through the sections, sub-sections and sub-sub-sections in order. Skip goes to the next section, sub-section or sub-sub-section at the same level in the hierarchy as the section, sub-section or sub-sub-section that you are currently reading. Up takes you up one level in the hierarchy and start gets you back here.
Please read the sections on customising and compilers before attempting to compile newmat.
| 1.1
Conditions of use 1.2 Description 1.3 Is this the library for you? 1.4 Other matrix libraries |
1.5
Where to find this library 1.6 How to contact the author 1.7 Change history 1.8 References |
There are no restrictions on the use of newmat except that I take no liability for any problems that may arise from this use.
I welcome its distribution as part of low cost CD-ROM collections.
You can use it in your commercial projects. However, if you distribute the source, please make it clear which parts are mine and that they are available essentially for free over the Internet.
Please understand that there may still be bugs and errors. Use at your own risk. I take no responsibility for any errors or omissions in this package or for any misfortune that may befall you or others as a result of its use.
Please report bugs to me at robert (at) statsresearch.co.nz [replace (at) by you-know-what-character in the email address].
When reporting a bug please tell me which C++ compiler you are using, and what version. Also give me details of your computer. And tell me which version of newmat (e.g. newmat03 or newmat04) you are using. Note any changes you have made to my code. If at all possible give me a piece of code illustrating the bug. See the problem report form.
Please do report bugs to me.
The package is intended for scientists and engineers who need to manipulate a variety of types of matrices using standard matrix operations. Emphasis is on the kind of operations needed in statistical calculations such as least squares, linear equation solve and eigenvalues.
It supports matrix types
Matrix rectangular matrix SquareMatrix square matrix nricMatrix for use with Numerical Recipes in C programs UpperTriangularMatrix LowerTriangularMatrix DiagonalMatrix SymmetricMatrix BandMatrix UpperBandMatrix upper triangular band matrix LowerBandMatrix lower triangular band matrix SymmetricBandMatrix RowVector derived from Matrix ColumnVector derived from Matrix IdentityMatrix diagonal matrix, elements have same value
Only one element type (float or double) is supported.
The package includes the operations *, +, -, Kronecker product, Schur product, concatenation, inverse, transpose, conversion between types, submatrix, determinant, Cholesky decomposition, QR decomposition, singular value decomposition, eigenvalues of a symmetric matrix, sorting, fast Fourier transform, printing and an interface with Numerical Recipes in C.
It is intended for matrices in the range 10 x 10 to the maximum size your machine will accommodate in a single array. The number of elements in an array cannot exceed the maximum size of an int. The package will work for very small matrices but becomes rather inefficient. Some of the factorisation functions are not (yet) optimised for paged memory and so become inefficient when used with very large matrices.
A lazy evaluation approach to evaluating matrix expressions is used to improve efficiency and reduce the use of temporary storage.
I have tested versions of the package on variety of compilers and platforms including Borland, Gnu, Microsoft and Sun. For more details see the section on compilers.
Do you
Then newmat may be the right matrix library for you.
For details of other C++ matrix libraries look at http://www.robertnz.net/cpp_site.html. Look at the section lists of libraries which gives the locations of several very comprehensive lists of matrix and other C++ libraries and at the section source code. Or just search on Google.
Robert Davies 16 Gloucester Street Wilton Wellington New Zealand Internet: robert(at)statsresearch.co.nz[replace (at) by you-know-what.]
Newmat11 - March, 2004:
Remove work-arounds for older compilers, Borland Builder 6 compatibility, SquareMatrix, load from array of ints, CrossProducts, Cholesky and QRZ update functions, swap functions, FFT2, access to arrays in traditional C functions, SimpleIntArray class, compatibility with Numerical Recipes in C++, speed-ups and bug-fixes.
Newmat10A - October, 2002:
Fix error in Kronecker product; fixes for Intel and GCC3 compilers.
Newmat10 - January, 2002:
Improve compatibility with GCC, fix errors in FFT and GenericMatrix, update simulated exceptions, maxima, minima, determinant, dot product and Frobenius norm functions, update make files for CC and GCC, faster FFT, A.ReSize(B), fix pointer arithmetic, << for loading data into rows, IdentityMatrix, Kronecker product, sort singular values.
Newmat09 - September, 1997:
Operator ==, !=, +=, -=, *=, /=, |=, &=. Follow new rules for for (int i; ... ) construct. Change Boolean, TRUE, FALSE to bool, true, false. Change ReDimension to ReSize. SubMatrix allows zero rows and columns. Scalar +, - or * matrix is OK. Simplify simulated exceptions. Fix non-linear programs for AT&T compilers. Dummy inequality operators. Improve internal row/column operations. Improve matrix LU decomposition. Improve sort. Reverse function. IsSingular function. Fast trig transforms. Namespace definitions.
Newmat08A - July, 1995:
Fix error in SVD.
Newmat08 - January, 1995:
Corrections to improve compatibility with Gnu, Watcom. Concatenation of matrices. Elementwise products. Option to use compilers supporting exceptions. Correction to exception module to allow global declarations of matrices. Fix problem with inverse of symmetric matrices. Fix divide-by-zero problem in SVD. Include new QR routines. Sum function. Non-linear optimisation. GenericMatrices.
Newmat07 - January, 1993
Minor corrections to improve compatibility with Zortech, Microsoft and Gnu. Correction to exception module. Additional FFT functions. Some minor increases in efficiency. Submatrices can now be used on RHS of =. Option for allowing C type subscripts. Method for loading short lists of numbers.
Newmat06 - December 1992:
Added band matrices; 'real' changed to 'Real' (to avoid potential conflict in complex class); Inject doesn't check for no loss of information; fixes for AT&T C++ version 3.0; real(A) becomes A.AsScalar(); CopyToMatrix becomes AsMatrix, etc; .c() is no longer required (to be deleted in next version); option for version 2.1 or later. Suffix for include files changed to .h; BOOL changed to Boolean (BOOL doesn't work in g++ v 2.0); modifications to allow for compilers that destroy temporaries very quickly; (Gnu users - see the section of compilers). Added CleanUp, LinearEquationSolver, primitive version of exceptions.
Newmat05 - June 1992:
For private release only
Newmat04 - December 1991:
Fix problem with G++1.40, some extra documentation
Newmat03 - November 1991:
Col and Cols become Column and Columns. Added Sort, SVD, Jacobi, Eigenvalues, FFT, real conversion of 1x1 matrix, Numerical Recipes in C interface, output operations, various scalar functions. Improved return from functions. Reorganised setting options in "include.hxx".
Newmat02 - July 1991:
Version with matrix row/column operations and numerous additional functions.
Matrix - October 1990:
Early version of package.
For references about Newmat see
I use .h as the suffix of definition files and .cpp as the suffix of C++ source files.
You will need to compile all the *.cpp files listed as program files in the files section to get the complete package. Ideally you should store the resulting object files as a library. The tmt*.cpp files are used for testing, example.cpp is an example and sl_ex.cpp, nl_ex.cpp and garch.cpp are examples of the non-linear solve and optimisation routines. A demonstration and test of the exception mechanism is in test_exc.cpp.
I include a number of make files for compiling the example and the test package. See the section on make files for details. But with the PC compilers, its pretty quick just to load all the files in the interactive environments by pointing and clicking.
Use the large or win32 console model when you are using a PC. Do not outline inline functions. You may need to increase the stack size.
Your source files that access the newmat will need to #include one or more of the following files.
| include.h | to access just the compiler options |
| newmat.h | to access just the main matrix library (includes include.h) |
| newmatap.h | to access the advanced matrix routines such as Cholesky decomposition, QR triangularisation etc (includes newmat.h) |
| newmatio.h | to access the output routines (includes newmat.h) You can use this only with compilers that support the standard input/output routines including manipulators |
| newmatnl.h | to access the non-linear optimisation routines (includes newmat.h) |
See the section on customising to see how to edit include.h for your environment and the section on compilers for any special problems with the compiler you are using.
I have included make files for CC, Borland 5.5, 5.6, Microsoft VC++6 and Gnu compilers. You can generate make files for a number of other compilers with my genmake utility. Make files provide a way of compiling your programs without using the IDE that comes with PC compilers. See the files section for details. See the example for how to use them. Leave out the target name to compile and link all my examples and test files.
I include make files for Borland 5.5, 5.6 and Microsoft VC++6 or 7. With the Borland compiler you will need to edit it to show where you have stored your Borland compiler. For make files for other compilers use my genmake utility. To compile my test and example programs use Borland 5.5 (Builder 5) use
make -f nm_b55.mak
or with Borland 5.6 (Builder 6) use
make -f nm_b56.mak
or with Microsoft VC++ 6 or 7 use
make -f nm_m6.mak
There are some more notes in the genmake documentation about using these make files.
The make file for the Unix CC compilers link a .cxx file to each .cpp file since some of these compilers do not recognise .cpp as a legitimate extension for a C++ file. I suggest you delete this part of the make file and, if necessary, rename the .cpp files to something your compiler recognises.
My make file for Gnu GCC on Unix systems is for use with gmake rather than make. I assume your compiler recognises the .cpp extension. Ordinary make works with it on the Sun but not the Silicon Graphics or HP machines. On Linux use make.
My make file for the CC compilers works with the ordinary make.
To compile everything with the CC compiler use
make -f nm_cc.mak
or for the gnu compiler use
make -f nm_gnu.mak
On some computers you will need to use gmake rather than make.
There is a line in the make file for CC rm -f $*.cxx. Some systems won't accept this line and you will need to delete it. In this case, if you have a bad compile and you are using my scheme for linking .cxx files, you will need to delete the .cxx file link generated by that compile before you can do the next one.
The file include.h sets a variety of options including several compiler dependent options. You may need to edit include.h to get the options you require. If you are using a compiler different from one I have worked with you may have to set up a new section in include.h appropriate for your compiler.
Borland, Turbo, Gnu, Microsoft and Watcom are recognised automatically. If none of these are recognised a default set of options is used. These are fine for AT&T, HPUX and Sun C++. If you using a compiler I don't know about you may have to write a new set of options.
There is an option in include.h for selecting whether you use compiler supported exceptions, simulated exceptions, or disable exceptions. I now set compiler supported exceptions as the default. Use the option for compiler supported exceptions if and only if you have set the option on your compiler to recognise exceptions. Disabling exceptions sometimes helps with compilers that are incompatible with my exception simulation scheme.
Activate the appropriate statement to make the element type float or double.
The option DO_FREE_CHECK is used for tracking memory leaks and normally should not be activated.
Activate SETUP_C_SUBSCRIPTS if you want to use traditional C style element access. Note that this does not change the starting point for indices when you are using round brackets for accessing elements or selecting submatrices. It does enable you to use C style square brackets. This also activates additional constructors for Matrix, ColumnVector and RowVector to simplify use with Numerical Recipes in C++.
Activate #define use_namespace if you want to use namespaces. Do this only if you are sure your compiler supports namespaces. If you do turn this option on, be prepared to turn it off again if the compiler reports inaccessible variables or the linker reports missing links.
Activate #define _STANDARD_ to use the standard names for the included files and to find the floating point precision data using the floating point standard. This will work only with the most recent compilers.
If you haven't defined _STANDARD_ and are using a compiler that include.h does not recognise and you want to pick up the floating point precision data from float.h then activate #define use_float_h. Otherwise the floating point precision data will be accessed from values.h. You may need to do this with computers from Digital, in particular.
|
2.4.1 AT&T 2.4.2 Borland 2.4.3 Gnu G++ 2.4.4 HPUX |
2.4.5 Intel 2.4.6 Microsoft 2.4.7 Sun 2.4.8 Watcom |
I have tested this library on a number of compilers. Here are the levels of success and any special considerations. In most cases I have chosen code that works under all the compilers I have access to, but I have had to include some specific work-arounds for some compilers. For the newest PC versions, I use a Pentium 4 computer running windows XP or Linux (Red Hat 9.0). The older compilers are tested on older computers. The Unix versions are on a Sun Sparc station. Thanks to Victoria University for access to the Sparc.
I have set up a block of code for each of the compilers in include.h. Turbo, Borland, Gnu, Microsoft and Watcom are recognised automatically. There is a default option that works for AT&T, Sun C++ and HPUX. So you don't have to make any changes for these compilers. Otherwise you may have to build your own set of options in include.h.
The AT&T compiler used to be available on a wide variety of Unix workstations. I don't know if anyone still uses it. However the AT&T options are the default if your compiler is not recognised.
AT&T C++ 2.1; 3.0.1 on a Sun: Previous versions worked on these compilers, which I no longer have access to.
In AT&T 2.1 you may get an error when you use an expression for the single argument when constructing a Vector or DiagonalMatrix or one of the Triangular Matrices. You need to evaluate the expression separately.
Borland Builder version 6: My tests have been on the personal version. See the notes for version 5. If you are compiling with a make file you can use nm_b56.mak as a model. You can set the newmat options to use namespace and the standard library. If you are compiling a GUI program you may need to comment out the line defining TypeDefException in include.h. I don't believe exceptions work completely correctly in either version 5 or version 6. However, this does not seem to be a problem with my use of them in newmat.
Borland Builder version 5: This works fine in console mode and no special editing of the source codes is required. I haven't tested it in GUI mode. You can set the newmat options to use namespace and the standard library. You should turn off the Borland option to use pre-compiled headers. There are notes on compiling with the IDE on my website. Alternatively you can use the nm_b55.mak make file.
Borland Builder version 4: I have successfully used this on older versions of newmat using the console wizard (menu item file/new - select new tab). Use compiler exceptions. Suppose you are compiling my test program tmt. Rename my main() function in tmt.cpp to my_main(). Rename tmt.cpp to tmt_main.cpp. Borland will generate a new file tmt.cpp containing their main() function. Put the line int my_main(); above this function and put return my_main(); into the body of main().
Borland compiler version 5.5: this is the free C++ compiler available from Borland's web site. I suggest you use the compiler supported exceptions and turn on standard in include.h. You can use the make file nm_b55.mak after editing to correct the file locations for your system.
Borland C++ 5.02: Use the large or 32 bit flat model. If you are not debugging, turn off the options that collect debugging information. Use my simulated exceptions.
When running my test program under ms-dos you may run out of memory. Either compile the test routine to run under easywin or use simulated exceptions rather than the built in exceptions.
If you can, upgrade to windows 95 or window NT and use the 32 bit console model.
If you are using the 16 bit large model, don't forget to keep all matrices less than 64K bytes in length (90x90 for a rectangular matrix if you are using double as your element type). Otherwise your program will crash without warning or explanation. You will need to break the tmt set of test files into several parts to get the program to fit into your computer and run without stack overflow.
One version of Borland had DBL_MIN incorrectly defined. If you are using an older version of Borland and are getting strange numerical errors in the test programs reinstate the commented out statements in precision.h.
You can generate make files for versions 5 with my genmake utility.
Borland C++ 3 and 4.
This version of Newmat is not compatible with these compilers.
Gnu G++ 2.95, 2.96, 3.1: This works OK. If you are using a much earlier version see if you can upgrade. You can't use standard with the 2.9X versions. The namespace option worked with 2.96 on Linux but not iwth 2.95 on the Sun. Standard is automatically turned on with the 3.X.
This version of Newmat is not compatible with versions 2.6 or earlier.
HP 9000 series HP-UX. I no longer have access to this compiler. Newmat09 worked without problems with the simulated exceptions; haven't tried the built-in exceptions.
With recent versions of the compiler you may get warning messages like Unsafe cast between pointers/references to incomplete classes. At present, I think these can be ignored.
Newmat works correctly with the Intel versions 5 and 8 C++ compiler for Windows and for versions 5 and 8 for Linux. (Not tested for the other versions). Standard is automatically turned on for the Linux versions. Note that the Intel compiler for Linux is free for non-commercial use.
See my web site for instructions how to work Microsoft's IDE.
Microsoft Visual C++ 7: This works OK. Note that all my tests have been in console mode. You can turn on my standard and namespace options.
Microsoft Visual C++ 6: Get the latest service pack. I have tried this in console mode and it seems to work satisfactorily. Use the compiler supported exceptions. You may be able to use the namespace and standard options. If you want to work under MFC you may need to #include "stdafx.h" at the beginning of each .cpp file (if you want precompiled headers).
Microsoft Visual C++ 5: I have tried this in console mode and it seems to work satisfactorily. There may be a problem with namespace (fixed by Service Pack 3?). Turn optimisation off. Use the compiler supported exceptions. If you want to work under MFC you may need to #include "stdafx.h" at the beginning of each .cpp file (if you want precompiled headers).
I doubt whether these will work.
Sun C++ (version 7): This seems to work fine with compiler supported exceptions. Sun C++ (version 5): There was a problem with exceptions. If you use my simulated exceptions the non-linear optimisation programs hang. If you use the compiler supported exceptions my tmt and test_exc programs crash. You should disable exceptions.
Watcom C++ (version 10a): this won't work as I no longer include the simulated Booleans. I haven't tried more recent versions of Watcom. (I can compile with the downloadable version of Watcom 11c but haven't found out how to link the files).
Newmat11 - if you are upgrading from earlier versions note the following:
Newmat10 includes new maxima, minima, determinant, dot product and Frobenius norm functions, a faster FFT, revised make files for GCC and CC compilers, several corrections, new ReSize function, IdentityMatrix and Kronecker Product. Singular values from SVD are sorted. The program files include a new file, newfft.cpp, so you will need to include this in the list of files in your IDE and make files. There is also a new test file tmtm.cpp. Pointer arithmetic now mostly meets requirements of standard. You can use << to load data into rows of a matrix. The default options in include.h have been changed. If you are updating from a beta version of newmat09 look through the next section as there were some late changes to newmat09.
If you are upgrading from newmat08 note the following:
If you are upgrading from newmat07 note the following:
If you are upgrading from newmat06 note the following:
If you are upgrading from newmat03 or newmat04 note the following
The current version is quite a bit longer that newmat04, so if you are almost out of space with newmat04, don't throw newmat04 away until you have checked your program will work under this version.
See the change history for other changes.
This section applies particularly to people using compiler supported exceptions rather than my simulated exceptions.
If newmat detects an error it will throw an exception. It is important that you catch this exception and print the error message. Otherwise you will get an unhelpful message like abnormal termination. I suggest you set up your main program like
#define WANT_STREAM // or #include <iostream>
#include "include.h" // or #include "newmat.h"
#include "myexcept.h"
main()
{
try
{
... your program here
}
// catch exceptions thrown by my programs
catch(Exception) { cout << Exception::what() << endl; }
// catch exceptions thrown by other people's programs
catch(...) { cout << "exception caught in main program" << endl; }
return 0;
}
If you are using a GUI version rather a console version of the program you will need to replace the cout statements by windows pop-up messages.
If you are using my simulated exceptions or have set the disable exceptions option in include.h then uncaught exceptions automatically print the error message generated by the exception so you can ignore this section. Alternatively use Try, Catch and CatchAll in place of try, catch and catch(...) in the preceding code.
See the section on exceptions for more information on the exception structure.
An example is given in example.cpp. This gives a simple linear regression example using five different algorithms. The correct output is given in example.txt. The program carries out a rough check that no memory is left allocated on the heap when it terminates. See the section on testing for a comment on the reliability of this check and the use of the DO_FREE_CHECK option.
I include a variety of make files. To compile the example use a command like
gmake -f nm_gnu.mak example (Gnu G++) gmake -f nm_cc.mak example (AT&T, HPUX, Sun) make -f nm_b55.mak example.exe (Borland C++ 5.5)
You can generate make make files for a number of other compilers with my genmake utility.
To compile all the example and test files use a command like
gmake -f nm_gnu.mak (Gnu G++)
The example uses io manipulators. It will not work with a compiler that does not support the standard io manipulators.
Other example files are nl_ex.cpp and garch.cpp for demonstrating the non-linear fitting routines, sl_ex for demonstrating the solve function and test_exc for demonstrating the exceptions.
The library package contains a comprehensive test program in the form of a series of files with names of the form tmt?.cxx. The files consist of a large number of matrix formulae all of which evaluate to zero (except the first one which is used to check that we are detecting non-zero matrices). The printout should state that it has found just one non-zero matrix.
The test program should be run with Real typedefed to double rather than float in include.h.
Make sure the C subscripts are enabled if you want to test these.
If you are carrying out some form of bounds checking, for example, with Borland's CodeGuard, then disable the testing of the Numerical Recipes in C interface. Activate the statement #define DONT_DO_NRIC in tmt.h.
Various versions of the make file (extension .mak) are included with the package. See the section on make files.
The program also allocates and deletes a large block and small block of memory before it starts the main testing and then at the end of the test. It then checks that the blocks of memory were allocated in the same place. If not, then one suspects that there has been a memory leak. i.e. a piece of memory has been allocated and not deleted.
This is not foolproof. Programs may allocate extra print buffers while the program is running. I have tried to overcome this by doing a print before I allocate the first memory block. Programs may allocate memory for different sized items in different places, or might not allocate items consecutively. Or they might mix the items with memory blocks from other programs. Nevertheless, I seem to get consistent answers from some of the compilers I work with, so I think this is a worthwhile test. The compilers that the test seems to work for include the Borland compilers, Microsoft VC++ 6 , Watcom 10a, and Gnu 2.96 for Linux.
If the DO_FREE_CHECK option in include.h is activated, the program checks that each new is balanced with exactly one delete. This provides a more definitive test of no memory leaks. There are additional statements in myexcept.cpp which can be activated to print out details of the memory being allocated and released.
I have included a facility for checking that each piece of code in the library is really exercised by the test routines. Each block of code in the main part of the library contains a word REPORT. newmat.h has a line defining REPORT that can be activated (deactivate the dummy version). This gives a printout of the number of times each of the REPORT statements in the .cpp files is accessed. Use a grep with line numbers to locate the lines on which REPORT occurs and compare these with the lines that the printout shows were actually accessed. One can then see which lines of code were not accessed.
This section lists parts of Newmat which users (including me) have found difficult or unnatural. Also see the newmat FAQ list on my website.
Invert, element access, matrix multiply etc causes the program to crash
Newmat throws an exception when it detects an error. This can cause a program crash unless the exception is caught with a catch statement. See catching exceptions.
1x1 matrix not automatically converted to a Real
Use the AsScalar() member function or the DotProduct() function to take the dot product of two vectors.
Constructors do not initialise elements
For example, Matrix A(4,5); does not initialise the elements of A. Use the statement A=0.0 to set the values to zero.
ReSize does not initialise elements
For example, A.ReSize(5,6); does not set the elements of A. See ReSize.
Setting Matrix to a scalar sets all the values
A(1,3) = 0.0; sets one element of a Matrix to zero. A = 0.0; sets all the elements to zero. This is very convenient but also a source of error that is hard to see if you wanted A(1,3) = 0.0; but left out the element details.
Symmetry not detected automatically
For example, SymmetricMatrix SM = A.t() * A; will fail. Use SymmetricMatrix SM; SM << A.t() * A;
Multiple multiplication may be inefficient
For example, A * B * X where A and B are matrices and X is a column vector is likely to be much slower than A * (B * X).
| Documentation | readme.txt | readme file |
| nm11.htm | documentation file | |
| add_time.pgn | image used by nm11.htm | |
| rbd.css | style sheet for nm11.htm | |
| Definition files | controlw.h | control word definition file |
| include.h | details of include files and options | |
| myexcept.h | general exception handler definitions | |
| newmat.h | main matrix class definition file | |
| newmatap.h | applications definition file | |
| newmatio.h | input/output definition file | |
| newmatnl.h | non-linear optimisation definition file | |
| newmatrc.h | row/column functions definition files | |
| newmatrm.h | rectangular matrix access definition files | |
| precisio.h | numerical precision constants | |
| solution.h | one dimensional solve definition file | |
| Program files | bandmat.cpp | band matrix routines |
| cholesky.cpp | Cholesky decomposition | |
| evalue.cpp | eigenvalues and eigenvector calculation | |
| fft.cpp | fast Fourier, trig. transforms | |
| hholder.cpp | QR routines | |
| jacobi.cpp | eigenvalues by the Jacobi method | |
| myexcept.cpp | general error and exception handler | |
| newfft.cpp | new fast Fourier transform | |
| newmat1.cpp | type manipulation routines | |
| newmat2.cpp | row and column manipulation functions | |
| newmat3.cpp | row and column access functions | |
| newmat4.cpp | constructors, resize, utilities | |
| newmat5.cpp | transpose, evaluate, matrix functions | |
| newmat6.cpp | operators, element access | |
| newmat7.cpp | invert, solve, binary operations | |
| newmat8.cpp | LU decomposition, scalar functions | |
| newmat9.cpp | output routines | |
| newmatex.cpp | matrix exception handler | |
| newmatnl.cpp | non-linear optimisation | |
| newmatrm.cpp | rectangular matrix access functions | |
| sort.cpp | sorting functions | |
| solution.cpp | one dimensional solve | |
| submat.cpp | submatrix functions | |
| svd.cpp | singular value decomposition | |
| Example files | example.cpp | example of use of package |
| example.txt | output from example | |
| sl_ex.cpp | example of OneDimSolve routine | |
| sl_ex.txt | output from example | |
| nl_ex.cpp | example of non-linear least squares | |
| nl_ex.txt | output from example | |
| garch.cpp | example of maximum-likelihood fit | |
| garch.dat | data file for garch.cpp | |
| garch.txt | output from example | |
| test_exc.cpp | demonstration exceptions | |
| test_exc.txt | output from test_exc.cpp | |
| Test files | tmt.h | header file for test files |
| tmt*.cpp | test files (see the section on testing) | |
| tmt.txt | output from test files | |
| Make files | nm_gnu.mak | make file for Gnu G++ |
| nm_cc.mak | make file for AT&T, Sun and HPUX | |
| nm_b55.mak | make file for Borland C++ 5.5 | |
| nm_b56.mak | make file for Borland Builder C++ 6 | |
| nm_m6.mak | make file for VC++ 6&7 | |
| nm_i5.mak | make file for Intel C++ under Windows | |
| nm_il5.mak | make file for Intel C++ under Linux | |
| newmat.lfl | library file list for use with genmake | |
| nm_targ.txt | target file list for use with genmake |
Copy and paste this to your editor; fill it out and email to robert (at) statsresearch.co.nz
But first look in my web page http://www.robertnz.net to see if the bug has already been reported.
Version: ............... newmat11 (28 March 2004) Your email address: .... Today's date: .......... Your machine: .......... Operating system: ...... Compiler & version: .... Compiler options (eg GUI or console)... Describe the problem - attach examples if possible: -----------------------------------------------------------
To construct an m x n matrix, A, (m and n are integers) use
Matrix A(m,n);
The SquareMatrix, UpperTriangularMatrix, LowerTriangularMatrix, SymmetricMatrix and DiagonalMatrix types are square. To construct an n x n matrix use, for example
SquareMatrix SQ(n);
UpperTriangularMatrix UT(n);
LowerTriangularMatrix LT(n);
SymmetricMatrix S(n);
DiagonalMatrix D(n);
Band matrices need to include bandwidth information in their constructors.
BandMatrix BM(n, lower, upper);
UpperBandMatrix UB(n, upper);
LowerBandMatrix LB(n, lower);
SymmetricBandMatrix SB(n, lower);
The integers upper and lower are the number of non-zero diagonals above and below the diagonal (excluding the diagonal) respectively.
The RowVector and ColumnVector types take just one argument in their constructors:
RowVector RV(n);
ColumnVector CV(n);
These constructors do not initialise the elements of the matrices. To set all the elements to zero use, for example,
Matrix A(m, n); A = 0.0;
The IdentityMatrix takes one argument in its constructor specifying its dimension.
IdentityMatrix I(n);
The value of the diagonal elements is set to 1 by default, but you can change this value as with other matrix types.
You can also construct vectors and matrices without specifying the dimension. For example
Matrix A;
In this case the dimension must be set by an assignment statement or a re-size statement.
You can also use a constructor to set a matrix equal to another matrix or matrix expression.
Matrix A = UT;
Matrix A = UT * LT;
Only conversions that don't lose information are supported - eg you cannot convert an upper triangular matrix into a diagonal matrix using =.
Elements are accessed by expressions of the form A(i,j) where i and j run from 1 to the appropriate dimension. Access elements of vectors with just one argument. Diagonal matrices can accept one or two subscripts.
This is different from the earliest version of the package in which the subscripts ran from 0 to one less than the appropriate dimension. Use A.element(i,j) if you want this earlier convention.
A(i,j) and A.element(i,j) can appear on either side of an = sign.
If you activate the #define SETUP_C_SUBSCRIPTS in include.h you can also access elements using the traditional C style notation. That is A[i][j] for matrices (except diagonal) and V[i] for vectors and diagonal matrices. The subscripts start at zero (i.e. like element) and there is no range checking. Because of the possibility of confusing V(i) and V[i], I suggest you do not activate this option unless you really want to use it.
Symmetric matrices are stored as lower triangular matrices. It is important to remember this if you are using the A[i][j] method of accessing elements. Make sure the first subscript is greater than or equal to the second subscript. However, if you are using the A(i,j) method the program will swap i and j if necessary; so it doesn't matter if you think of the storage as being in the upper triangle (but it does matter in some other situations such as when entering data).
The IdentityMatrix type does not support element access.
For interfacing with traditional C functions that involve one and two dimensional arrays see accessing C functions.
The operator = is used for copying matrices, converting matrices, or evaluating expressions. For example
A = B; A = L; A = L * U;
Only conversions that don't lose information are supported. The dimensions of the matrix on the left hand side are adjusted to those of the matrix or expression on the right hand side. Elements on the right hand side which are not present on the left hand side are set to zero.
The operator << can be used in place of = where it is permissible for information to be lost.
For example
SymmetricMatrix S; Matrix A;
......
S << A.t() * A;
is acceptable whereas
S = A.t() * A; // error
will cause a runtime error since the package does not (yet?) recognise A.t()*A as symmetric.
Note that you can not use << with constructors. For example
SymmetricMatrix S << A.t() * A; // error
does not work.
Also note that << cannot be used to load values from a full matrix into a band matrix, since it will be unable to determine the bandwidth of the band matrix.
A third copy routine is used in a similar role to =. Use
A.Inject(D);
to copy the elements of D to the corresponding elements of A but leave the elements of A unchanged if there is no corresponding element of D (the = operator would set them to 0). This is useful, for example, for setting the diagonal elements of a matrix without disturbing the rest of the matrix. Unlike = and <<, Inject does not reset the dimensions of A, which must match those of D. Inject does not test for no loss of information.
You cannot replace D by a matrix expression. The effect of Inject(D) depends on the type of D. If D is an expression it might not be obvious to the user what type it would have. So I thought it best to disallow expressions.
Inject can be used for loading values from a regular matrix into a band matrix. (Don't forget to zero any elements of the left hand side that will not be set by the loading operation).
Both << and Inject can be used with submatrix expressions on the left hand side. See the section on submatrices.
To set the elements of a matrix to a scalar use operator =
Real r; int m,n;
......
Matrix A(m,n); A = r;
To swap the values in two matrices A and B use one of the
following expressionsA.swap(B);
swap(A,B);The matrices A and B must be of the same type. This can be any of the matrix types listed in the section on constructors, CroutMatrix, BandLUMatrix or GenericMatrix. Swap works by switching pointers and does not do any actual copying of the main data arrays.
Notes:
You can load the elements of a matrix from an array:
Matrix A(3,2);
Real a[] = { 11,12,21,22,31,33 };
A << a;
or
Matrix A(3,2);
int a[] = { 11,12,21,22,31,33 };
A << a;
This construction does not check that the numbers of elements match correctly. This version of << can be used with submatrices on the left hand side. It is not defined for band matrices.
Alternatively you can enter short lists using a sequence of numbers separated by << .
Matrix A(3,2);
A << 11 << 12
<< 21 << 22
<< 31 << 32;
This does check for the correct total number of entries, although the message for there being insufficient numbers in the list may be delayed until the end of the block or the next use of this construction. This does not work for band matrices or for long lists. It does work for submatrices if the submatrix is a single complete row. For example
Matrix A(3,2);
A.Row(1) << 11 << 12;
A.Row(2) << 21 << 22;
A.Row(3) << 31 << 32;
Load only values that are actually stored in the matrix. For example
SymmetricMatrix S(2);
S.Row(1) << 11;
S.Row(2) << 21 << 22;
Try to restrict this way of loading data to numbers. You can include expressions, but these must not call a function which includes the same construction.
Remember that matrices are stored by rows and that symmetric matrices are stored as lower triangular matrices when using these methods to enter data.
The package supports unary operations
X = -A; // change sign of elements
X = A.t(); // transpose
X = A.i(); // inverse (of square matrix A)
X = A.Reverse(); // reverse order of elements of vector
// or matrix (not band matrix)
The package supports binary operations
X = A + B; // matrix addition
X = A - B; // matrix subtraction
X = A * B; // matrix multiplication
X = A.i() * B; // equation solve (square matrix A)
X = A | B; // concatenate horizontally (concatenate the rows)
X = A & B; // concatenate vertically (concatenate the columns)
X = SP(A, B); // elementwise product of A and B (Schur product)
X = KP(A, B); // Kronecker product of A and B
X = CrossProduct(A, B); // vector cross product - see notes
X = CrossProductRows(A, B); // cross product of rows
X = CrossProductColumns(A, B); // cross product of columns
bool b = A == B; // test whether A and B are equal
bool b = A != B; // ! (A == B)
A += B; // A = A + B;
A -= B; // A = A - B;
A *= B; // A = A * B;
A |= B; // A = A | B;
A &= B; // A = A & B;
<, >, <=, >= // included for compatibility with STL - see notes
Notes:
Remember that the product of symmetric matrices is not necessarily symmetric so the following code will not run:
SymmetricMatrix A, B; .... put values in A, B .... SymmetricMatrix C = A * B; // run time error
Use instead
Matrix C = A * B;
or, if you know the product will be symmetric, use
SymmetricMatrix C; C << A * B;
The following expressions multiply the elements of a matrix A by a scalar f: A * f or f * A . Likewise one can divide the elements of a matrix A by a scalar f: A / f .
The expressions A + f and A - f add or subtract a rectangular matrix of the same dimension as A with elements equal to f to or from the matrix A .
The expression f + A is an alternative to A + f. The expression f - A subtracts matrix A from a rectangular matrix of the same dimension as A and with elements equal to f .
The expression A += f replaces A by A + f. Operators -=, *=, /= are similarly defined.
This page describes functions returning the values associated with the size and shape of matrices. The following pages describe other scalar matrix functions.
int m = A.nrows(); // number of rows
int m = A.Nrows(); // number of rows
int n = A.ncols(); // number of columns
int n = A.Ncols(); // number of columns
MatrixType mt = A.Type(); // type of matrix
Real* s = A.data(); // pointer to array of elements
const Real* s = A.data(); // pointer to array of elements
// where A is const
const Real* s = A.const_data(); // pointer to array of elements
Real* s = A.Store(); // pointer to array of elements
int l = A.size(); // length of array of elements
int l = A.Storage(); // length of array of elements
MatrixBandWidth mbw = A.BandWidth(); // upper and lower bandwidths
MatrixType mt = A.Type() returns the type of a matrix. Use (char*)mt to get a string (UT, LT, Rect, Sym, Diag, Band, UB, LB, Crout, BndLU) showing the type (Vector types are returned as Rect).
MatrixBandWidth has member functions Upper() and Lower() for finding the upper and lower bandwidths (number of diagonals above and below the diagonal, both zero for a diagonal matrix). For non-band matrices -1 is returned for both these values.
This page describes functions for finding the maximum and minimum elements of a matrix.
int i, j;
Real mv = A.MaximumAbsoluteValue(); // maximum of absolute values
Real mv = A.MinimumAbsoluteValue(); // minimum of absolute values
Real mv = A.Maximum(); // maximum value
Real mv = A.Minimum(); // minimum value
Real mv = A.MaximumAbsoluteValue1(i); // maximum of absolute values
Real mv = A.MinimumAbsoluteValue1(i); // minimum of absolute values
Real mv = A.Maximum1(i); // maximum value
Real mv = A.Minimum1(i); // minimum value
Real mv = A.MaximumAbsoluteValue2(i,j);// maximum of absolute values
Real mv = A.MinimumAbsoluteValue2(i,j);// minimum of absolute values
Real mv = A.Maximum2(i,j); // maximum value
Real mv = A.Minimum2(i,j); // minimum value
All these functions throw an exception if A has no rows or no columns.
The versions A.MaximumAbsoluteValue1(i), etc return the location of the extreme element in a RowVector, ColumnVector or DiagonalMatrix. The versions A.MaximumAbsoluteValue2(i,j), etc return the row and column numbers of the extreme element. If the extreme value occurs more than once the location of the last one is given.
The versions MaximumAbsoluteValue(A), MinimumAbsoluteValue(A), Maximum(A), Minimum(A) can be used in place of A.MaximumAbsoluteValue(), A.MinimumAbsoluteValue(), A.Maximum(), A.Minimum().
Real r = A.AsScalar(); // value of 1x1 matrix
Real ssq = A.SumSquare(); // sum of squares of elements
Real sav = A.SumAbsoluteValue(); // sum of absolute values
Real s = A.Sum(); // sum of values
Real norm = A.Norm1(); // maximum of sum of absolute
values of elements of a column
Real norm = A.NormInfinity(); // maximum of sum of absolute
values of elements of a row
Real norm = A.NormFrobenius(); // square root of sum of squares
// of the elements
Real t = A.Trace(); // trace
Real d = A.Determinant(); // determinant
LogAndSign ld = A.LogDeterminant(); // log of determinant
bool z = A.IsZero(); // test all elements zero
bool s = A.IsSingular(); // A is a CroutMatrix or
BandLUMatrix
Real s = DotProduct(A, B); // dot product of A and B
// interpreted as vectors
A.LogDeterminant() returns a value of type LogAndSign. If ld is of type LogAndSign use
ld.Value() to get the value of the determinant
ld.Sign() to get the sign of the determinant (values 1, 0, -1)
ld.LogValue() to get the log of the absolute value.
Note that the direct use of the function Determinant() will often cause a floating point overflow exception.
A.IsZero() returns Boolean value true if the matrix A has all elements equal to 0.0.
IsSingular is defined only for CroutMatrix and BandLUMatrix. It returns true if one of the diagonal elements of the LU decomposition is exactly zero.
DotProduct(const Matrix& A,const Matrix& B) converts both of the arguments to rectangular matrices, checks that they have the same number of elements and then calculates the first element of A * first element of B + second element of A * second element of B + ... ignoring the row/column structure of A and B. It is primarily intended for the situation where A and B are row or column vectors.
The versions Sum(A), SumSquare(A), SumAbsoluteValue(A), Trace(A), LogDeterminant(A), Determinant(A), Norm1(A), NormInfinity(A), NormFrobenius(A) can be used in place of A.Sum(), A.SumSquare(), A.SumAbsoluteValue(), A.Trace(), A.LogDeterminant(), A.Norm1(), A.NormInfinity(), A.NormFrobenius().
A.SubMatrix(fr,lr,fc,lc)
This selects a submatrix from A. The arguments fr,lr,fc,lc are the first row, last row, first column, last column of the submatrix with the numbering beginning at 1.
I allow lr = fr-1 or lc = fc-1 or to indicate that a matrix of zero rows or columns is to be returned.
A submatrix command may be used in any matrix expression or on the left hand side of =, << or Inject. Inject does not check no information loss. You can also use the construction
Real c; .... A.SubMatrix(fr,lr,fc,lc) = c;
to set a submatrix equal to a constant.
The following are variants of SubMatrix:
A.SymSubMatrix(f,l) // This assumes fr=fc and lr=lc.
A.Rows(f,l) // select rows
A.Row(f) // select single row
A.Columns(f,l) // select columns
A.Column(f) // select single column
In each case f and l mean the first and last row or column to be selected (starting at 1).
I allow l = f-1 to indicate that a matrix of zero rows or columns is to be returned.
If SubMatrix or its variant occurs on the right hand side of an = or << or within an expression think of its type as follows
A.SubMatrix(fr,lr,fc,lc) If A is RowVector or
ColumnVector then same type
otherwise type Matrix
A.SymSubMatrix(f,l) Same type as A
A.Rows(f,l) Type Matrix
A.Row(f) Type RowVector
A.Columns(f,l) Type Matrix
A.Column(f) Type ColumnVector
If SubMatrix or its variant appears on the left hand side of = or << , think of its type being Matrix. Thus L.Row(1) where L is LowerTriangularMatrix expects L.Ncols() elements even though it will use only one of them. If you are using = the program will check for no loss of data.
A SubMatrix can appear on the left-hand side of += or -= with a matrix expression on the right-hand side. It can also appear on the left-hand side of +=, -=, *= or /= with a Real on the right-hand side. In each case there must be no loss of information.
The Row version can appear on the left hand side of << for loading literal data into a row. Load only the number of elements that are actually going to be stored in memory.
Do not use the += and -= operations with a submatrix of a SymmetricMatrix or BandSymmetricMatrix on the LHS and a Real on the RHS.
You can't pass a submatrix (or any of its variants) as a reference non-constant matrix in a function argument. For example, the following will not work:
void YourFunction(Matrix& A); ... Matrix B(10,10); YourFunction(B.SubMatrix(1,5,1,5)) // won't compile
If you are are using the submatrix facility to build a matrix from a small number of components, consider instead using the concatenation operators.
The following operations change the dimensions of a matrix. The values of the elements are lost.
A.ReSize(nrows,ncols); // for type Matrix or nricMatrix
A.ReSize(n); // for all other types, except Band
A.ReSize(n,lower,upper); // for BandMatrix
A.ReSize(n,lower); // for LowerBandMatrix
A.ReSize(n,upper); // for UpperBandMatrix
A.ReSize(n,lower); // for SymmetricBandMatrix
A.ReSize(B); // set dims to those of B
Use A.CleanUp() to set the dimensions of A to zero and release all the heap memory.
A.ReSize(B) sets the dimensions of A to those of a matrix B. This includes the band-width in the case of a band matrix. It is an error for A to be a band matrix and B not a band matrix (or diagonal matrix).
Remember that ReSize destroys values. If you want to ReSize, but keep the values in the bit that is left use something like
ColumnVector V(100); ... // load values V = V.Rows(1,50); // to get first 50 values.
If you want to extend a matrix or vector use something like
ColumnVector V(50);
... // load values
{ V.Release(); ColumnVector X=V; V.ReSize(100); V.Rows(1,50)=X; }
// V now length 100
The following functions interpret the elements of a matrix (stored row by row) to be a vector or matrix of a different type. Actual copying is usually avoided where these occur as part of a more complicated expression.
A.AsRow()
A.AsColumn()
A.AsDiagonal()
A.AsMatrix(nrows,ncols)
A.AsScalar()
The expression A.AsScalar() is used to convert a 1 x 1 matrix to a scalar.
To solve the matrix equation Ay = b where A is a square matrix of equation coefficients, y is a column vector of values to be solved for, and b is a column vector, use the code
int n = something
Matrix A(n,n); ColumnVector b(n);
... put values in A and b
ColumnVector y = A.i() * b; // solves matrix equation
The following notes are for the case where you want to solve more than one matrix equation with different values of b but the same A. Or where you want to solve a matrix equation and also find the determinant of A. In these cases you probably want to avoid repeating the LU decomposition of A for each solve or determinant calculation.
If A is a square or symmetric matrix use
CroutMatrix X = A; // carries out LU decomposition
Matrix AP = X.i()*P; Matrix AQ = X.i()*Q;
LogAndSign ld = X.LogDeterminant();
rather than
Matrix AP = A.i()*P; Matrix AQ = A.i()*Q;
LogAndSign ld = A.LogDeterminant();
since each operation will repeat the LU decomposition.
If A is a BandMatrix or a SymmetricBandMatrix begin with
BandLUMatrix X = A; // carries out LU decomposition
A CroutMatrix or a BandLUMatrix can't be manipulated or copied. Use references as an alternative to copying.
Alternatively use
LinearEquationSolver X = A;
This will choose the most appropriate decomposition of A. That is, the band form if A is banded; the Crout decomposition if A is square or symmetric and no decomposition if A is triangular or diagonal.
The package does not support delayed copy. Several strategies are required to prevent unnecessary matrix copies.
Where a matrix is called as a function argument use a constant reference. For example
YourFunction(const Matrix& A)
rather than
YourFunction(Matrix A)
Skip the rest of this section on your first reading.
A second place where it is desirable to avoid unnecessary copies is when a function is returning a matrix. Matrices can be returned from a function with the return command as you would expect. However these may incur one and possibly two copyings of the matrix. To avoid this use the following instructions.
Make your function of type ReturnMatrix . Then precede the return statement with a Release statement (or a ReleaseAndDelete statement if the matrix was created with new). For example
ReturnMatrix MakeAMatrix()
{
Matrix A; // or any other matrix type
......
A.Release(); return A;
}
or
ReturnMatrix MakeAMatrix()
{
Matrix* m = new Matrix;
......
m->ReleaseAndDelete(); return *m;
}
If your compiler objects to this code, replace the return statements with
return A.ForReturn();
or
return m->ForReturn();
If you are using AT&T C++ you may wish to replace return A; by return (ReturnMatrix)A; to avoid a warning message; but this will give a runtime error with Gnu. (You can't please everyone.)
Do not forget to make the function of type ReturnMatrix; otherwise you may get incomprehensible run-time errors.
You can also use .Release() or ->ReleaseAndDelete() to allow a matrix expression to recycle space. Suppose you call
A.Release();
just before A is used just once in an expression. Then the memory used by A is either returned to the system or reused in the expression. In either case, A's memory is destroyed. This procedure can be used to improve efficiency and reduce the use of memory.
Use ->ReleaseAndDelete for matrices created by new if you want to completely delete the matrix after it is accessed.
The package tends to be not very efficient for dealing with matrices with short rows. This is because some administration is required for accessing rows for a variety of types of matrices. To reduce the administration a special multiply routine is used for rectangular matrices in place of the generic one. Where operations can be done without reference to the individual rows (such as adding matrices of the same type) appropriate routines are used.
When you are using small matrices (say smaller than 10 x 10) you may find it faster to use rectangular matrices rather than the triangular or symmetric ones.
To print a matrix use an expression like
Matrix A; ...... cout << setw(10) << setprecision(5) << A;
This will work only with systems that support the standard input/output routines including manipulators. You need to #include the files iostream.h, iomanip.h, newmatio.h in your C++ source files that use this facility. The files iostream.h, iomanip.h will be included automatically if you include the statement #define WANT_STREAM at the beginning of your source file. So you can begin your file with either
#define WANT_STREAM #include "newmatio.h"
or
#include <iostream.h> #include <iomanip.h> #include "newmatio.h"
The present version of this routine is useful only for matrices small enough to fit within a page or screen width.
To print several vectors or matrices in columns use a concatenation operator:
ColumnVector A, B; ..... cout << setw(10) << setprecision(5) << (A | B);
Skip this section on your first reading.
If you want to work with a matrix of unknown type, say in a function. You can construct a matrix of type GenericMatrix. Eg
Matrix A; ..... // put some values in A GenericMatrix GM = A;
A GenericMatrix matrix can be used anywhere where a matrix expression can be used and also on the left hand side of an =. You can pass any type of matrix (excluding the Crout and BandLUMatrix types) to a const GenericMatrix& argument in a function. However most scalar functions including nrows(), ncols(), Type() and element access do not work with it. Nor does the ReturnMatrix construct. swap does work with objects of type GenericMatrix. See also the paragraph on LinearEquationSolver.
An alternative and less flexible approach is to use BaseMatrix or GeneralMatrix.
Suppose you wish to write a function which accesses a matrix of unknown type including expressions (eg A*B). Then use a layout similar to the following:
void YourFunction(BaseMatrix& X)
{
GeneralMatrix* gm = X.Evaluate(); // evaluate an expression
// if necessary
........ // operations on *gm
gm->tDelete(); // delete *gm if a temporary
}
See, as an example, the definitions of operator<< in newmat9.cpp.
Under certain circumstances; particularly where X is to be used just once in an expression you can leave out the Evaluate() statement and the corresponding tDelete(). Just use X in the expression.
If you know YourFunction will never have to handle a formula as its argument you could also use
void YourFunction(const GeneralMatrix& X)
{
........ // operations on X
}
Do not try to construct a GeneralMatrix or BaseMatrix.
Suppose S is symmetric and positive definite. Then there exists a unique lower triangular matrix L such that L * L.t() = S. To calculate this use
SymmetricMatrix S;
......
LowerTriangularMatrix L = Cholesky(S);
If S is a symmetric band matrix then L is a band matrix and an alternative procedure is provided for carrying out the decomposition:
SymmetricBandMatrix S;
......
LowerBandMatrix L = Cholesky(S);
See section 3.32 on updating a Cholesky
decomposition.
This is a variant on the usual QR transformation.
Start with matrix (dimensions shown to left and below the matrix)
/ 0 0 \ s
\ X Y / n
s t
Our version of the QR decomposition multiplies this matrix by an orthogonal matrix Q to get
/ U M \ s
\ 0 Z / n
s t
where U is upper triangular (the R of the QR transform). That is
Q / 0 0 \ = / U M \
\ X Y / \ 0 Z /
This is good for solving least squares problems: choose b (matrix or column vector) to minimise the sum of the squares of the elements of
Y - X*b
Then choose b = U.i()*M; The residuals Y - X*b are in Z.
This is the usual QR transformation applied to the matrix X with the square zero matrix concatenated on top of it. It gives the same triangular matrix as the QR transform applied directly to X and generally seems to work in the same way as the usual QR transform. However it fits into the matrix package better and also gives us the residuals directly. It turns out to be essentially a modified Gram-Schmidt decomposition.
Two routines are provided in newmat:
QRZ(X, U);
replaces X by orthogonal columns and forms U.
QRZ(X, Y, M);
uses X from the first routine, replaces Y by Z and forms M.
The are also two routines QRZT(X, L) and QRZT(X, Y, M) which do the same decomposition on the transposes of all these matrices. QRZT replaces the routines HHDecompose in earlier versions of newmat. HHDecompose is still defined but just calls QRZT.
For an example of the use of this decomposition see the file example.cpp.
See section 3.32 on updating U.
Alternatively to update U or L with a new block of data, X, one can use
UpdateQRZ(X, U);
or
UpdateQRZT(X, L);
Notes on UpdateQRZ and UpdateQRZT:
The singular value decomposition of an m x n Matrix A (where m >= n) is a decomposition
A = U * D * V.t()
where U is m x n with U.t() * U equalling the identity, D is an n x n DiagonalMatrix and V is an n x n orthogonal matrix (type Matrix in Newmat).
Singular value decompositions are useful for understanding the structure of ill-conditioned matrices, solving least squares problems, and for finding the eigenvalues of A.t() * A.
To calculate the singular value decomposition of A (with m >= n) use one of
SVD(A, D, U, V); // U = A is OK
SVD(A, D);
SVD(A, D, U); // U = A is OK
SVD(A, D, U, false); // U (can = A) for workspace only
SVD(A, D, U, V, false); // U (can = A) for workspace only
where A, U and V are of type Matrix and D is a DiagonalMatrix. The values of A are not changed unless A is also inserted as the third argument.
The elements of D are sorted in descending order.
Remember that the SVD decomposition is not completely unique. The signs of the elements in a column of U may be reversed if the signs in the corresponding column in V are reversed. If a number of the singular values are identical one can apply an orthogonal transformation to the corresponding columns of U and the corresponding columns of V.
An eigenvalue decomposition of a SymmetricMatrix A is a decomposition
A = V * D * V.t()
where V is an orthogonal matrix (type Matrix in Newmat) and D is a DiagonalMatrix.
Eigenvalue analyses are used in a wide variety of engineering, statistical and other mathematical analyses.
The package includes two algorithms: Jacobi and Householder. The first is extremely reliable but much slower than the second.
The code is adapted from routines in Handbook for Automatic Computation, Vol II, Linear Algebra by Wilkinson and Reinsch, published by Springer Verlag.
Jacobi(A,D,S,V); // A, S symmetric; S is workspace,
// S = A is OK; V is a matrix
Jacobi(A,D); // A symmetric
Jacobi(A,D,S); // A, S symmetric; S is workspace,
// S = A is OK
Jacobi(A,D,V); // A symmetric; V is a matrix
EigenValues(A,D); // A symmetric
EigenValues(A,D,S); // A, S symmetric; S is for back
// transforming, S = A is OK
EigenValues(A,D,V); // A symmetric; V is a matrix
where A, S are of type SymmetricMatrix, D is of type DiagonalMatrix and V is of type Matrix. The values of A are not changed unless A is also inserted as the third argument. If you need eigenvectors use one of the forms with matrix V. The eigenvectors are returned as the columns of V.
The elements of D are sorted in ascending order.
Remember that an eigenvalue decomposition is not completely unique - see the comments about the SVD decomposition.
To sort the values in a matrix or vector, A, (in general this operation makes sense only for vectors and diagonal matrices) use one of
SortAscending(A);
SortDescending(A);
I use the quicksort algorithm. The algorithm is similar to that in Sedgewick's algorithms in C++. If the sort seems to be failing (as quicksort can do) an exception is thrown.
You will get incorrect results if you try to sort a band matrix - but why would you want to sort a band matrix?
FFT(X, Y, F, G); // F=X and G=Y are OK
where X, Y, F, G are ColumnVectors. X and Y are the real and imaginary input vectors; F and G are the real and imaginary output vectors. The lengths of X and Y must be equal and should be the product of numbers less than about 10 for fast execution.
The formula is
n-1
h[k] = SUM z[j] exp (-2 pi i jk/n)
j=0
where z[j] is stored complex and stored in X(j+1) and Y(j+1). Likewise h[k] is complex and stored in F(k+1) and G(k+1). The fast Fourier algorithm takes order n log(n) operations (for good values of n) rather than n**2 that straight evaluation (see the file tmtf.cpp) takes.
I use one of two methods:
Related functions
FFTI(F, G, X, Y); // X=F and Y=G are OK RealFFT(X, F, G); RealFFTI(F, G, X);
FFTI is the inverse transform for FFT. RealFFT is for the case when the input vector is real, that is Y = 0. I assume the length of X, denoted by n, is even. That is, n must be divisible by 2. The program sets the lengths of F and G to n/2 + 1. RealFFTI is the inverse of RealFFT.
See also the section on fast trigonometric transforms.
There are also two dimensional versions
FFT2(X, Y, F, G); // F=X and G=Y are OK FFT2I(F, G, X, Y); // inverse, X=F and Y=G are OK
where X, Y, F, G are of type Matrix. X and Y are the real and imaginary input matrices; F and G are the real and imaginary output matrices. The dimensions of Y must be the same as those of X and should be the product of numbers less than about 10 for fast execution.
The formula is
m-1 n-1
h[p,q] = SUM SUM z[j,k] exp (-2 pi i (jp/m + kq/n))
j=0 k=0
where z[j,k] is stored complex and stored in X(j+1,k+1) and Y(j+1,k+1) and X and Y have dimension m x n. Likewise h[p,q] is complex and stored in F(p+1,q+1) and G(p+1,q+1).
These are the sin and cosine transforms as defined by Charles Van Loan (1992) in Computational frameworks for the fast Fourier transform published by SIAM. See page 229. Some other authors use slightly different conventions. All the functions call the fast Fourier transforms and require an even transform length, denoted by m in these notes. That is, m must be divisible by 2. As with the FFT m should be the product of numbers less than about 10 for fast execution.
The functions I define are
DCT(U,V); // U, V are ColumnVectors, length m+1 DCT_inverse(V,U); // inverse of DCT DST(U,V); // U, V are ColumnVectors, length m+1 DST_inverse(V,U); // inverse of DST DCT_II(U,V); // U, V are ColumnVectors, length m DCT_II_inverse(V,U); // inverse of DCT_II DST_II(U,V); // U, V are ColumnVectors, length m DST_II_inverse(V,U); // inverse of DST_II
where the first argument is the input and the second argument is the output. V = U is OK. The length of the output ColumnVector is set by the functions.
Here are the formulae:
m-1 k
v[k] = u[0]/2 + SUM { u[j] cos (pi jk/m) } + (-) u[m]/2
j=1
for k = 0...m, where u[j] and v[k] are stored in U(j+1) and V(k+1).
m-1
v[k] = SUM { u[j] sin (pi jk/m) }
j=1
for k = 1...(m-1), where u[j] and v[k] are stored in U(j+1) and V(k+1)and where u[0] and u[m] are ignored and v[0] and v[m] are set to zero. For the inverse function v[0] and v[m] are ignored and u[0] and u[m] are set to zero.
m-1
v[k] = SUM { u[j] cos (pi (j+1/2)k/m) }
j=0
for k = 0...(m-1), where u[j] and v[k] are stored in U(j+1) and V(k+1).
m
v[k] = SUM { u[j] sin (pi (j-1/2)k/m) }
j=1
for k = 1...m, where u[j] and v[k] are stored in U(j) and V(k).
Note that the relationship between the subscripts in the formulae and those used in newmat is different for DST_II (and DST_II_inverse).
This section refers to Numerical Recipes in C. This section is not relevant to Numerical Recipes in C++. I'll put a note on the website soon on how to interface with Numerical Recipes in C++.
This package can be used with the vectors and matrices defined in Numerical Recipes in C. You need to edit the routines in Numerical Recipes so that the elements are of the same type as used in this package. Eg replace float by double, vector by dvector and matrix by dmatrix, etc. You may need to edit the function definitions to use the version acceptable to your compiler (if you are using the first edition of NRIC). You may need to enclose the code from Numerical Recipes in extern "C" { ... }. You will also need to include the matrix and vector utility routines.
Then any vector in Numerical Recipes with subscripts starting from 1 in a function call can be accessed by a RowVector, ColumnVector or DiagonalMatrix in the present package. Similarly any matrix with subscripts starting from 1 can be accessed by an nricMatrix in the present package. The class nricMatrix is derived from Matrix and can be used in place of Matrix. In each case, if you wish to refer to a RowVector, ColumnVector, DiagonalMatrix or nricMatrix X in an function from Numerical Recipes, use X.nric() in the function call.
Numerical Recipes cannot change the dimensions of a matrix or vector. So matrices or vectors must be correctly dimensioned before a Numerical Recipes routine is called.
For example
SymmetricMatrix B(44);
..... // load values into B
nricMatrix BX = B; // copy values to an nricMatrix
DiagonalMatrix D(44); // Matrices for output
nricMatrix V(44,44); // correctly dimensioned
int nrot;
jacobi(BX.nric(),44,D.nric(),V.nric(),&nrot);
// jacobi from NRIC
cout << D; // print eigenvalues
Here is the class structure for exceptions:
Exception
Logic_error
ProgramException miscellaneous matrix error
IndexException index out of bounds
VectorException unable to convert matrix to vector
NotSquareException matrix is not square (invert, solve)
SubMatrixDimensionException out of bounds index of submatrix
IncompatibleDimensionsException (multiply, add etc)
NotDefinedException operation not defined (eg <)
CannotBuildException copying a matrix where copy is undefined
InternalException probably an error in newmat
Runtime_error
NPDException matrix not positive definite (Cholesky)
ConvergenceException no convergence (e-values, non-linear, sort)
SingularException matrix is singular (invert, solve)
SolutionException no convergence in solution routine
OverflowException floating point overflow
Bad_alloc out of space (new fails)
I have attempted to mimic the exception class structure in the C++ standard library, by defining the Logic_error and Runtime_error classes.
Suppose you have edited include.h to use my simulated exceptions or to disable exceptions. If there is no catch statement or exceptions are disabled then my Terminate() function in myexcept.h is called when you throw an exception. This prints out an error message, the dimensions and types of the matrices involved, the name of the routine detecting the exception, and any other information set by the Tracer class. Also see the section on error messages for additional notes on the messages generated by the exceptions.
You can also print this information in a catch clause by printing Exception::what().
If you are using compiler supported exceptions then see the section on catching exceptions.
See the file test_exc.cpp as an example of catching an exception and printing the error message.
The 08 version of newmat defined a member function void SetAction(int) to help customise the action when an exception is called. This has been deleted in the 09 and 10 versions. Now include an instruction such as cout << Exception::what() << endl; in the Catch or CatchAll block to determine the action.
The library includes the alternatives of using the inbuilt exceptions provided by a compiler, simulating exceptions, or disabling exceptions. See customising for selecting the correct exception option.
The rest of this section describes my partial simulation of exceptions for compilers which do not support C++ exceptions. I use Carlos Vidal's article in the September 1992 C Users Journal as a starting point.
Newmat does a partial clean up of memory following throwing an exception - see the next section. However, the present version will leave a little heap memory unrecovered under some circumstances. I would not expect this to be a major problem, but it is something that needs to be sorted out.
The functions/macros I define are Try, Throw, Catch, CatchAll and CatchAndThrow. Try, Throw, Catch and CatchAll correspond to try, throw, catch and catch(...) in the C++ standard. A list of Catch clauses must be terminated by either CatchAll or CatchAndThrow but not both. Throw takes an Exception as an argument or takes no argument (for passing on an exception). I do not have a version of Throw for specifying which exceptions a function might throw. Catch takes an exception class name as an argument; CatchAll and CatchAndThrow don't have any arguments. Try, Catch and CatchAll must be followed by blocks enclosed in curly brackets.
I have added another macro ReThrow to mean a rethrow, Throw(). This was necessary to enable the package to be compatible with both my exception package and C++ exceptions.
If you want to throw an exception, use a statement like
Throw(Exception("Error message\n"));
It is important to have the exception declaration in the Throw statement, rather than as a separate statement.
All exception classes must be derived from the class, Exception, defined in newmat and can contain only static variables. See the examples in newmat if you want to define additional exceptions.
Note that the simulation exception mechanism does not work if you define arrays of matrices.
This section is about the simulated exceptions used in newmat. It is irrelevant if you are using the exceptions built into a compiler or have set the disable-exceptions option.
The simulated exception mechanisms in newmat are based on the C functions setjmp and longjmp. These functions do not call destructors so can lead to garbage being left on the heap. (I refer to memory allocated by new as heap memory). For example, when you call
Matrix A(20,30);
a small amount of space is used on the stack containing the row and column dimensions of the matrix and 600 doubles are allocated on the heap for the actual values of the matrix. At the end of the block in which A is declared, the destructor for A is called and the 600 doubles are freed. The locations on the stack are freed as part of the normal operations of the stack. If you leave the block using a longjmp command those 600 doubles will not be freed and will occupy space until the program terminates.
To overcome this problem newmat keeps a list of all the currently declared matrices and its exception mechanism will return heap memory when you do a Throw and Catch.
However it will not return heap memory from objects from other packages.
If you want the mechanism to work with another class you will have to do four things:
public:
void* operator new(size_t size)
{ do_not_link=true; void* t = ::operator new(size); return t; }
void operator delete(void* t) { ::operator delete(t); }
X(const X&);
Note that the function CleanUp() does somewhat the same duties as the destructor. However CleanUp() has to do the cleaning for the class you are working with and also the classes it is derived from. So it will often be wrong to use exactly the same code for both CleanUp() and the destructor or to define your destructor as a call to CleanUp().
Files solution.h, solution.cpp contain a class for solving for x in y = f(x) where x is a one-dimensional continuous monotonic function. This is not a matrix thing at all but is included because it is a useful program and because it is a simpler version of the coding technique used in the non-linear least squares.
Files newmatnl.h, newmatnl.cpp contain a series of classes for non-linear least squares and maximum likelihood. These classes work on very well-behaved functions but need upgrading for less well-behaved functions. I haven't followed the usual practice of inflating the values of the diagonal elements of the matrix of second derivatives. I originally thought I could avoid this if my program had a good line search. But this was wrong and when I use this program on all but the most well-behaved problems I run the fit first with the diagonal elements inflated by a factor of 2 to 5 and the critical value for the stopping criterion set to something like 50. Then rerun with with no inflation factor and critical value 0.0001.
Documentation for both of these is in the definition files. Simple examples are in sl_ex.cpp, nl_ex.cpp and garch.cpp.
The standard template library (STL) is the set of container templates (vector, deque, list etc) defined by the C++ standards committee. Newmat is intended to be compatible with the STL in the sense that you can store matrices in the standard containers. I have defined == and inequality operators which seem to be required by some versions of the STL.
If you want to use the container classes with Newmat please note
Namespace is used to avoid name clashes between different libraries. I have included the namespace capability. Activate the line #define use_namespace in include.h. Then include either the statement
using namespace NEWMAT;
at the beginning of any file that needs to access the newmat library or
using namespace RBD_LIBRARIES;
at the beginning of any file that needs to access all my libraries.
This works correctly with Borland C++ version 5 and Builder 5 and 6.
Microsoft Visual C++ version 5 works in my example and test files, but fails with apparently insignificant changes (it may be more reliable if you have applied service pack 3). If you #include "newmatap.h", but no other newmat include file, then also #include "newmatio.h". It seems to work with Microsoft Visual C++ version 6 if you have applied at least service pack 2.
My use of namespace does not work with Gnu g++ version 2.8.1 but does work with versions 3.x.
I have defined the following namespaces:
Suppose X is matrix and U has been formed with either
SymmetricMatrix A; A << X.t() * X; UpperTriangularMatrix U = Cholesky(A).t();
or
UpperTriangularMatrix U; QRZ(X, U);
Now suppose we want to append an extra row to X or delete a row from X or rearrange the columns of X. The functions described here allow you to update U without recalculating it.
UpdateCholesky(U, x); // x is a RowVector DowndateCholesky(U, x); // x is a RowVector RightCircularUpdateCholesky(U, j, k); // j and k are integers LeftCircularUpdateCholesky(U, j, k); // j and k are integers
UpdateCholesky carries out the modification of U corresponding to appending an extra row x to X.
DowndateCholesky carries out the modification corresponding to removing a row x from X. A ProgramException exception is thrown if the modification fails.
RightCircularUpdateCholesky supposes that columns j,j+1,...k of X are replaced by columns k,j,j+1,...k-1.
LeftCircularUpdateCholesky supposes that columns j,j+1,...k of X are replaced by columns j+1,...k,j.
These functions are based on a contribution from Nick Bennett of Schlumberger-Doll Research. See also the LINPACK User's Guide, Chapter 10, Dongarra et. al., SIAM, Philadelphia, 1979.
Where you want to append a number of new rows consider using the update routine in section 3.20.
You have a C function that uses one and two dimensional arrays as vectors and matrices. You want to access it from Newmat.
One dimensional arrays are easy. Set up a ColumnVector, RowVector or DiagonalMatrix with the correct dimension and where the function has a double* argument enter X.data() where X denotes the ColumnVector, RowVector or DiagonalMatrix. (I am assuming you have left Real being typedefed as a double). If you have a const double* argument use X.const_data().
You can't do this with two dimensional arrays where you have a double** argument. Newmat includes classes RealStarStar and ConstRealStarStar for this situation. To access a Matrix A with from a function c_function(double** a) use either
c_function(RealStarStar(A));
or
RealStarStar a(A); c_function(a);
If the argument is const double** use ConstRealStarStar.
The Matrix A must be correctly dimensioned and must not be resized or set equal to another matrix between setting up the RealStarStar object and its use in the function. Also the following construction will not work
double** a = RealStarStar(A); // wrong c_function(a);
since the RealStarStar structure will have been destroyed before you get to the second line.
This is primarily for use within Newmat. You can set up a simple array of integers with the SimpleIntArray class. Here are the descriptions of the constructors and functions.
| SimpleIntArray A; | Constructs int array of length zero |
| SimpleIntArray A(n); | Constructs int array of length n - individual elements are not initialised |
| A = i; | sets values to i where i is an int variable |
| A = B; | sets values to those of B where B is a SimpleIntArray, change size if necessary. |
| n = A.size(); | return the length of A |
| A.resize(n); | change the length of A, don't keep values |
| A.resize(n, true); | change the length of A, do keep values |
| int x = A[i]; | element access; i runs from 0 to n-1 |
| int* d = A.data(); | get beginning of data array |
| const int* d = A.data(); | get beginning of data array as const int* if A is const |
| const int* d = A.const_data(); | get beginning of data array as const int* |
Most error messages are self-explanatory. The message gives the size of the matrices involved. Matrix types are referred to by the following codes:
Matrix or vector Rect Symmetric matrix Sym Band matrix Band Symmetric band matrix SmBnd Lower triangular matrix LT Lower triangular band matrix LwBnd Upper triangular matrix UT Upper triangular band matrix UpBnd Diagonal matrix Diag Crout matrix (LU matrix) Crout Band LU matrix BndLU
Other codes should not occur.
See the section on exceptions for more details on the structure of the exception classes.
I have defined a class Tracer that is intended to help locate the place where an error has occurred. At the beginning of a function I suggest you include a statement like
Tracer tr("name");
where name is the name of the function. This name will be printed as part of the error message, if an exception occurs in that function, or in a function called from that function. You can change the name as you proceed through a function with the ReName function
tr.ReName("new name");
if, for example, you want to track progress through the function.
I describe some of the ideas behind this package, some of the decisions that I needed to make and give some details about the way it works. You don't need to read this part of the documentation in order to use the package.
It isn't obvious what is the best way of going about structuring a matrix package. I don't think you can figure this out with thought experiments. Different people have to try out different approaches. And someone else may have to figure out which is best. Or, more likely, the ultimate packages will lift some ideas from each of a variety of trial packages. So, I don't claim my package is an ultimate package, but simply a trial of a number of ideas. The following pages give some background on these ideas.
A library like newmat needs to balance safety, usability and efficiency.
By safety, I mean getting the right answer, and not causing crashes or damage to the computer system.
By usability, I mean being easy to learn and use, including not being too complicated, being intuitive, saving the users' time, being nice to use.
Efficiency means minimising the use of computer memory and time.
In the early days of computers the emphasis was on efficiency. But computer power gets cheaper and cheaper, halving in price every 18 months. On the other hand the unaided human brain is probably not a lot better than it was 100,000 years ago! So we should expect the balance to shift to put more emphasis on safety and usability and a little less on efficiency. So I don't mind if my programs are a little less efficient than programs written in pure C (or Fortran) if I gain substantially in safety and usability. But I would mind if they were a lot less efficient.
Second reason for putting extra emphasis on safety and usability is the way I and, I suspect, most other users actually use newmat. Most completed programs are used only a few times. Some result is required for a client, paper or thesis. The program is developed and tested, the result is obtained, and the program archived. Of course bits of the program will be recycled for the next project. But it may be less usual for the same program to be run over and over again. So the cost, computer time + people time, is in the development time and often, much less in the actual time to run the final program. So good use of people time, especially during development is really important. This means you need highly usable libraries.
So if you are dealing with matrices, you want the good interface that I have tried to provide in newmat, and, of course, reliable methods underneath it.
Of course, efficiency is still important. We often want to run the biggest problem our computer will handle and often a little bigger. The C++ language almost lets us have both worlds. We can define a reasonably good interface, and get good efficiency in the use of the computer.
We can imagine the black box model of a newmat. Suppose the inside is hidden but can be accessed by the methods described in the reference section. Then the interface is reasonably consistent and intuitive. Matrices can be accessed and manipulated in much the same way as doubles or ints in regular C. All accesses are checked. It is most unlikely that an incorrect index will crash the system. In general, users do not need to use pointers, so one shouldn't get pointers pointing into space. And, hopefully, you will get simpler code with less errors.
There are some exceptions to this. In particular, the C-like subscripts are not checked for validity. They give faster access but with a lower level of safety.
Then there is the data() function which takes you to the data array within a matrix. This takes you right inside the black box. But this is what you have to use if you are writing, for example, a new matrix factorisation, and require fast access to the data array. I have tried to write code to simplify access to the interior of a rectangular matrix, see file newmatrm.cpp, but I don't regard this as very successful, as yet, and have not included it in the documentation. Ideally we should have improved versions of this code for each of the major types of matrix. But, in reality, most of my matrix factorisations are written in what is basically the C language with very little C++.
So our box is not very black. You have a choice of how far you penetrate. On the outside you have a good level of safety, but in some cases efficiency is compromised a little. If you penetrate inside the box safety is reduced but you can get better efficiency.
This section looks at the performance on newmat for simple sums, comparing it with C code and with a simple array program.
The following table lists the time (in seconds) for carrying out the operations X=A+B;, X=A+B+C;, X=A+B+C+D;, X=A+B+C+D+E; where X,A,B,C,D,E are of type ColumnVector, with a variety of programs. I am using Microsoft VC++, version 6 in console mode under windows 2000 on a PC with a 1 ghz Pentium III and 512 mbytes of memory.
length iters. newmat C C-res. subs. array
X = A + B
2 5000000 27.8 0.3 8.8 1.9 9.5
20 500000 3.0 0.3 1.1 1.9 1.2
200 50000 0.5 0.3 0.4 1.9 0.3
2000 5000 0.4 0.3 0.4 2.0 1.0
20000 500 4.5 4.5 4.5 6.7 4.4
200000 50 5.2 4.7 5.5 5.8 5.2
X = A + B + C
2 5000000 36.6 0.4 8.9 2.5 12.2
20 500000 4.0 0.4 1.2 2.5 1.6
200 50000 0.8 0.3 0.5 2.5 0.5
2000 5000 3.6 4.4 4.6 9.0 4.4
20000 500 6.8 5.4 5.4 9.6 6.8
200000 50 8.6 6.0 6.7 7.1 8.6
X = A + B + C + D
2 5000000 44.0 0.7 9.3 3.1 14.6
20 500000 4.9 0.6 1.5 3.1 1.9
200 50000 1.0 0.6 0.8 3.2 0.8
2000 5000 5.6 6.6 6.8 11.5 5.9
20000 500 9.0 6.7 6.8 11.0 8.5
200000 50 11.9 7.1 7.9 9.5 12.0
X = A + B + C + D + E
2 5000000 50.6 1.0 9.5 3.8 17.1
20 500000 5.7 0.8 1.7 3.9 2.4
200 50000 1.3 0.9 1.0 3.9 1.0
2000 5000 7.0 8.3 8.2 13.8 7.1
20000 500 11.5 8.1 8.4 13.2 11.0
200000 50 15.2 8.7 9.5 12.4 15.4
I have graphed the results and included rather more array lengths.
The first column gives the lengths of the arrays, the second the number of iterations and the remaining columns the total time required in seconds. If the only thing that consumed time was the double precision addition then the numbers within each block of the table would be the same. The summation is repeated 5 times within each loop, for example:
for (i=1; i<=m; ++i)
{
X1 = A1+B1+C1; X2 = A2+B2+C2; X3 = A3+B3+C3;
X4 = A4+B4+C4; X5 = A5+B5+C5;
}
The column labelled newmat is using the standard newmat add. The column labelled C uses the usual C method: while (j1--) *x1++ = *a1++ + *b1++; . The following column also includes an X.ReSize() in the outer loop to correspond to the reassignment of memory that newmat would do. In the next column the calculation is using the usual C style for loop and accessing the elements using newmat subscripts such as A(i). The final column is the time taken by a simple array package. This uses an alternative method for avoiding temporaries and unnecessary copies that does not involve runtime tests. It does its sums in blocks of 4 and copies in blocks of 8 in the same way that newmat does.
Here are my conclusions.
In summary: for the situation considered here, newmat is doing very well for large ColumnVectors, even for sums with several terms, but not so well for shorter ColumnVectors.
The newmat library is for the manipulation of matrices, including the standard operations such as multiplication as understood by numerical analysts, engineers and mathematicians.
A matrix is a two dimensional array of numbers. However, very special operations such as matrix multiplication are defined specifically for matrices. This means that a matrix library, as I understand the term, is different from a general array library. Here are some contrasting properties.
| Feature | Matrix library | Array library |
|---|---|---|
| Expressions | Matrix expressions; * means matrix multiply; inverse function | Arithmetic operations, if supported, mean elementwise combination of arrays |
| Element access | Access to the elements of a matrix | High speed access to elements directly and perhaps with iterators |
| Elementary functions | For example: determinant, trace | Matrix multiplication as a function |
| Advanced functions | For example: eigenvalue analysis | |
| Element types | Real and possibly complex | Wide range - real, integer, string etc |
| Types | Rectangular, symmetric, diagonal, etc | One, two and three dimensional arrays, at least |
Both types of library need to support access to sub-matrices or sub-arrays, have good efficiency and storage management, and graceful exit for errors. In both cases, we probably need two versions, one optimised for large matrices or arrays and one for small matrices or arrays.
It may be possible to amalgamate the two sets of requirements to some extent. However newmat is definitely oriented towards the matrix library set.
Even within the bounds set by the requirements of a matrix library there is a substantial opportunity for variation between what different matrix packages might provide. It is not possible to build a matrix package that will meet everyone's requirements. In many cases if you put in one facility, you impose overheads on everyone using the package. This both in storage required for the program and in efficiency. Likewise a package that is optimised towards handling large matrices is likely to become less efficient for very small matrices where the administration time for the matrix may become significant compared with the time to carry out the operations. It is better to provide a variety of packages (hopefully compatible) so that most users can find one that meets their requirements. This package is intended to be one of these packages; but not all of them.
Since my background is in statistical methods, this package is oriented towards the kinds things you need for statistical analyses.
Now looking at some specific questions.
A matrix library may target small matrices (say 3 x 3), or medium sized matrices, or very large matrices.
A library targeting very small matrices will seek to minimise administration. A library for medium sized or very large matrices can spend more time on administration in order to conserve space or optimise the evaluation of expressions. A library for very large matrices will need to pay special attention to storage and numerical properties. This library is designed for medium sized matrices. This means it is worth introducing some optimisations, but I don't have to worry about setting up some form of virtual memory.
As well as the usual rectangular matrices, matrices occurring repeatedly in numerical calculations are upper and lower triangular matrices, symmetric matrices and diagonal matrices. This is particularly the case in calculations involving least squares and eigenvalue calculations. So as a first stage these were the types I decided to include.
It is also necessary to have types row vector and column vector. In a matrix package, in contrast to an array package, it is necessary to have both these types since they behave differently in matrix expressions. The vector types can be derived for the rectangular matrix type, so having them does not greatly increase the complexity of the package.
The problem with having several matrix types is the number of versions of the binary operators one needs. If one has 5 distinct matrix types then a simple library will need 25 versions of each of the binary operators. In fact, we can evade this problem, but at the cost of some complexity.
Ideally we would allow element types double, float, complex and int, at least. It might be reasonably easy, using templates or equivalent, to provide a library which could handle a variety of element types. However, as soon as one starts implementing the binary operators between matrices with different element types, again one gets an explosion in the number of operations one needs to consider. At the present time the compilers I deal with are not up to handling this problem with templates. (Of course, when I started writing newmat there were no templates). But even when the compilers do meet the specifications of the draft standard, writing a matrix package that allows for a variety of element types using the template mechanism is going to be very difficult. I am inclined to use templates in an array library but not in a matrix library.
Hence I decided to implement only one element type. But the user can decide whether this is float or double. The package assumes elements are of type Real. The user typedefs Real to float or double.
It might also be worth including symmetric and triangular matrices with extra precision elements (double or long double) to be used for storage only and with a minimum of operations defined. These would be used for accumulating the results of sums of squares and product matrices or multi-stage QR decompositions.
I want to be able to write matrix expressions the way I would on paper. So if I want to multiply two matrices and then add the transpose of a third one I can write something like X = A * B + C.t();. I want this expression to be evaluated with close to the same efficiency as a hand-coded version. This is not so much of a problem with expressions including a multiply since the multiply will dominate the time. However, it is not so easy to achieve with expressions with just + and -.
A second requirement is that temporary matrices generated during the evaluation of an expression are destroyed as quickly as possible.
A desirable feature is that a certain amount of intelligence be displayed in the evaluation of an expression. For example, in the expression X = A.i() * B; where i() denotes inverse, it would be desirable if the inverse wasn't explicitly calculated.
How are classes and public member functions to be named? As a general rule I have spelt identifiers out in full with individual words being capitalised. For example UpperTriangularMatrix. If you don't like this you can #define or typedef shorter names. This convention means you can select an abbreviation scheme that makes sense to you.
Exceptions to the general rule are the functions for transpose and inverse. To make matrix expressions more like the corresponding mathematical formulae, I have used the single letter abbreviations, t() and i().
In mathematical work matrix subscripts usually start at one. In C, array subscripts start at zero. In Fortran, they start at one. Possibilities for this package were to make them start at 0 or 1 or be arbitrary.
Alternatively one could specify an index set for indexing the rows and columns of a matrix. One would be able to add or multiply matrices only if the appropriate row and column index sets were identical.
In fact, I adopted the simpler convention of making the rows and columns of a matrix be indexed by an integer starting at one, following the traditional convention. In an earlier version of the package I had them starting at zero, but even I was getting mixed up when trying to use this earlier package. So I reverted to the more usual notation and started at 1.
We want to be able to use the notation A(i,j) to specify the (i,j)-th element of a matrix. This is the way mathematicians expect to address the elements of matrices. I consider the notation A[i][j] totally alien. However I include this as an option to help people converting from C.
There are two ways of working out the address of A(i,j). One is using a dope vector which contains the first address of each row. Alternatively you can calculate the address using the formula appropriate for the structure of A. I use this second approach. It is probably slower, but saves worrying about an extra bit of storage.
The other question is whether to check for i and j being in range. I do carry out this check following years of experience with both systems that do and systems that don't do this check. I would hope that the routines I supply with this package will reduce your need to access elements of matrices so speed of access is not a high priority.
Iterators are an alternative way of providing fast access to the elements of an array or matrix when they are to be accessed sequentially. They need to be customised for each type of matrix. I have not implemented iterators in this package, although some iterator like functions are used internally for some row and column functions.
To understand how newmat stores matrices you need to know a little bit about the heap and stack.
The data values of variables or objects in a C++ program are stored in either of two sections of memory called the stack and the heap. Sometimes there is more than one heap to cater for different sized variables.
If you declare an automatic variable
int x;
then the value of x is stored on the stack. As you declare more variables the stack gets bigger. When you exit a block (i.e a section of code delimited by curly brackets {...}) the memory used by the automatic variables declared in the block is released and the stack shrinks.
When you declare a variable with new, for example,
int* y = new int;
the pointer y is stored on the stack but the value it is pointing to is stored on the heap. Memory on the heap is not released until the program explicitly does this with a delete statement
delete *y;
or the program exits.
On the stack, variables and objects are is always added to the end of the stack and are removed in the reverse order to that in which they are added - that is the last on will be the first off. This is not the case with the heap, where the variables and objects can be removed in any order. So one can get alternating pieces of used and unused memory. When a new variable or object is declared on the heap the system needs to search for piece of unused memory large enough to hold it. This means that storing on the heap will usually be a slower process than storing on the stack. There is also likely to be waste space on the heap because of gaps between the used blocks of memory that are too small for the next object you want to store on the heap. There is also the possibility of wasting space if you forget to remove a variable or object on the heap even though you have finished using it. However, the stack is usually limited to holding small objects with size known at compile time. Large objects, objects whose size you don't know at compile time, and objects that you want to persist after the end of the block need to be stored on the heap.
In C++, the constructor/destructor system enables one to build complicated objects such as matrices that behave as automatic variables stored on the stack, so the programmer doesn't have to worry about deleting them at the end of the block, but which really utilise the heap for storing their data.
Each matrix object contains the basic information such as the number of rows and columns, the amount of memory used, a status variable and a pointer to the data array which is on the heap. So if you declare a matrix
Matrix A(1000,1000);
there is an small amount of memory used on the stack for storing the numbers of rows and columns, the amount of memory used, the status variable and the pointer together with 1,000,000 Real locations stored on the heap. When you exit the block in which A is declared, the heap memory used by A is automatically returned to the system, as well as the memory used on the stack.
Of course, if you use new to declare a matrix
Matrix* B = new Matrix(1000,1000);
both the information about the size and the actual data are stored on heap and not deleted until the program exits or you do an explicit delete:
delete *B;
If you carry out an assignment with = or << or do a resize() the data array currently associated with a matrix is destroyed and a new array generated. For example
Matrix A(1000,1000); Matrix B(50, 50); ... put some values in B A = B;
At the last step the heap memory associated with A is returned to the system and a new block of heap memory is assigned to contain the new values. This happens even if there is no change in the amount of memory required.
The elements of the matrix are stored as a single array. Alternatives would have been to store each row as a separate array or a set of adjacent rows as a separate array. The present solution simplifies the program but limits the size of matrices in 16 bit PCs that have a 64k byte limit on the size of arrays (I don't use the huge keyword). The large arrays may also cause problems for memory management in smaller machines. [The 16 bit PC problem has largely gone away but it was a problem when much of newmat was written. Now, occasionally I run into the 32 bit PC problem.]
In Fortran two dimensional arrays are stored by column. In most other systems they are stored by row. I have followed this later convention. This makes it easier to interface with other packages written in C but harder to interface with those written in Fortran. This may have been a wrong decision. Most work on the efficient manipulation of large matrices is being done in Fortran. It would have been easier to use this work if I had adopted the Fortran convention.
An alternative would be to store the elements by mid-sized rectangular blocks. This might impose less strain on memory management when one needs to access both rows and columns.
Symmetric matrices are stored as lower triangular matrices. The decision was pretty arbitrary, but it does slightly simplify the Cholesky decomposition program.
Consider the instruction
X = A + B + C;
To evaluate this a simple program will add A to B putting the total in a temporary T1. Then it will add T1 to C creating another temporary T2 which will be copied into X. T1 and T2 will sit around till the end of the execution of the statement and perhaps of the block. It would be faster if the program recognised that T1 was temporary and stored the sum of T1 and C back into T1 instead of creating T2 and then avoided the final copy by just assigning the contents of T1 to X rather than copying. In this case there will be no temporaries requiring deletion. (More precisely there will be a header to be deleted but no contents).
For an instruction like
X = (A * B) + (C * D);
we can't easily avoid one temporary being left over, so we would like this temporary deleted as quickly as possible.
I provide the functionality for doing all this by attaching a status variable to each matrix. This indicates if the matrix is temporary so that its memory is available for recycling or deleting. Any matrix operation checks the status variables of the matrices it is working with and recycles or deletes any temporary memory.
An alternative or additional approach would be to use reference counting and delayed copying - also known as copy on write. If a program requests a matrix to be copied, the copy is delayed until an instruction is executed which modifies the memory of either the original matrix or the copy. If the original matrix is deleted before either matrix is modified, in effect, the values of the original matrix are transferred to the copy without any actual copying taking place. This solves the difficult problem of returning an object from a function without copying and saves the unnecessary copying in the previous examples.
There are downsides to the delayed copying approach. Typically, for delayed copying one uses a structure like the following:
Matrix
|
+------> Array Object
| |
| +------> Data array
| |
| +------- Counter
|
+------ Dimension information
where the arrows denote a pointer to a data structure. If one wants to access the Data array one will need to track through two pointers. If one is going to write, one will have to check whether one needs to copy first. This is not important when one is going to access the whole array, say, for a add operation. But if one wants to access just a single element, then it imposes a significant additional overhead on that operation. Any subscript operation would need to check whether an update was required - even read since it is hard for the compiler to tell whether a subscript access is a read or write.
Some matrix libraries don't bother to do this. So if you write A = B; and then modify an element of one of A or B, then the same element of the other is also modified. I don't think this is acceptable behaviour.
Delayed copy does not provide the additional functionality of my approach but I suppose it would be possible to have both delayed copy and tagging temporaries.
My approach does not automatically avoid all copying. In particular, you need use a special technique to return a matrix from a function without copying.
Modern computers work faster if one accesses memory by running through contiguous locations rather than by jumping around all over the place. Newmat stores matrices by rows so that algorithms that access memory by running along rows will tend to work faster than one that runs down columns. A number of the algorithms used in Newmat were developed before this was an issue and so are not as efficient as possible.
I have gradually upgrading the algorithms to access memory by rows. The following table shows the current status of this process.
| Function | Contiguous memory access | Comment |
|---|---|---|
| Add, subtract | Yes | |
| Multiply | Yes | |
| Concatenate | Yes | |
| Transpose | No | |
| Invert and solve | Yes | Mostly |
| Cholesky | Yes | |
| QRZ, QRZT | Yes | |
| SVD | No | |
| Jacobi | No | Not an issue; used only for smaller matrices |
| Eigenvalues | No | |
| Sort | Yes | Quick-sort is naturally good |
| FFT | ? | Could be improved? |
This is now all rather out of date. With Pentiums, at least, the important requirement for speed seems to be to minimise transfers between the RAM memory and the on-chip memory. There isn't much you can do about add and subtract, but there lots of possibilities for some of the other operations.
Consider the instruction
X = B - X;
A simple program will subtract X from B, store the result in a temporary T1 and copy T1 into X. It would be faster if the program recognised that the result could be stored directly into X. This would happen automatically if the program could look at the instruction first and mark X as temporary.
C programmers would expect to avoid the same problem with
X = X - B;
by using an operator -=
X -= B;
However this is an unnatural notation for non C users and it may be nicer to write X = X - B; and know that the program will carry out the simplification.
Another example where this intelligent analysis of an instruction is helpful is in
X = A.i() * B;
where i() denotes inverse. Numerical analysts know it is inefficient to evaluate this expression by carrying out the inverse operation and then the multiply. Yet it is a convenient way of writing the instruction. It would be helpful if the program recognised this expression and carried out the more appropriate approach.
I regard this interpretation of A.i() * B as just providing a convenient notation. The objective is not primarily to correct the errors of people who are unaware of the inefficiency of A.i() * B if interpreted literally.
There is a third reason for the two-stage evaluation of expressions and this is probably the most important one. In C++ it is quite hard to return an expression from a function such as (*, + etc) without a copy. This is particularly the case when an assignment (=) is involved. The mechanism described here provides one way for avoiding this in matrix expressions.
The C++ standard (section 12.8/15) allows the compiler to optimise away the copy when returning an object from a function (but there will still be one copy is an assignment (=) is involved). This means special handling of returns from a function is less important when a modern optimising compiler is being used.
To carry out this intelligent analysis of an instruction matrix expressions are evaluated in two stages. In the the first stage a tree representation of the expression is formed. For example (A+B)*C is represented by a tree
*
/ \
+ C
/ \
A B
Rather than adding A and B the + operator yields an object of a class AddedMatrix which is just a pair of pointers to A and B. Then the * operator yields a MultipliedMatrix which is a pair of pointers to the AddedMatrix and C. The tree is examined for any simplifications and then evaluated recursively.
Further possibilities not yet included are to recognise A.t()*A and A.t()+A as symmetric or to improve the efficiency of evaluation of expressions like A+B+C, A*B*C, A*B.t() (t() denotes transpose).
One of the disadvantages of the two-stage approach is that the types of matrix expressions are determined at run-time. So the compiler will not detect errors of the type
Matrix M; DiagonalMatrix D; ....; D = M;
We don't allow conversions using = when information would be lost. Such errors will be detected when the statement is executed.
The package attempts to solve the problem of the large number of versions of the binary operations required when one has a variety of types.
With n types of matrices the binary operations will each require n-squared separate algorithms. Some reduction in the number may be possible by carrying out conversions. However, the situation rapidly becomes impossible with more than 4 or 5 types. Doug Lea told me that it was possible to avoid this problem. I don't know what his solution is. Here's mine.
Each matrix type includes routines for extracting individual rows or columns. I assume a row or column consists of a sequence of zeros, a sequence of stored values and then another sequence of zeros. Only a single algorithm is then required for each binary operation. The rows can be located very quickly since most of the matrices are stored row by row. Columns must be copied and so the access is somewhat slower. As far as possible my algorithms access the matrices by row.
There is another approach. Each of the matrix types defined in this package can be set up so both rows and columns have their elements at equal intervals provided we are prepared to store the rows and columns in up to three chunks. With such an approach one could write a single "generic" algorithm for each of multiply and add. This would be a reasonable alternative to my approach.
I provide several algorithms for operations like + . If one is adding two matrices of the same type then there is no need to access the individual rows or columns and a faster general algorithm is appropriate.
Generally the method works well. However symmetric matrices are not always handled very efficiently (yet) since complete rows are not stored explicitly.
The original version of the package did not use this access by row or column method and provided the multitude of algorithms for the combination of different matrix types. The code file length turned out to be just a little longer than the present one when providing the same facilities with 5 distinct types of matrices. It would have been very difficult to increase the number of matrix types in the original version. Apparently 4 to 5 types is about the break even point for switching to the approach adopted in the present package.
However it must also be admitted that there is a substantial overhead in the approach adopted in the present package for small matrices. The test program developed for the original version of the package takes 30 to 50% longer to run with the current version (though there may be some other reasons for this). This is for matrices in the range 6x6 to 10x10.
To try to improve the situation a little I do provide an ordinary matrix multiplication routine for the case when all the matrices involved are rectangular.
Versions before version 5 of newmat did not work correctly with Gnu C++ (version 5 or earlier). This was because the tree structure used to represent a matrix expression was set up on the stack. Early versions of Gnu C++ destroyed temporary structures as soon as the function that accesses them finished. To overcome this problem, there was an option to store the temporaries forming the tree structure on the heap (created with new) and to delete them explicitly. Now that the C++ standards committee has said that temporary structures should not be destroyed before a statement finishes, I have deleted this option.
The program needs to be able to work out the class of the result of a matrix expression. This is to check that a conversion is legal or to determine the class of an intermediate result. To assist with this, a class MatrixType is defined. Operators +, -, *, >= are defined to calculate the types of the results of expressions or to check that conversions are legal.
Early versions of newmat stored the types of the results of operations in a table. So, for example, if you multiplied an UpperTriangularMatrix by a LowerTriangularMatrix, newmat would look up the table and see that the result was of type Matrix. With this approach the exploding number of operations problem recurred although not as seriously as when code had to be written for each pair of types. But there was always the suspicion that somewhere, there was an error in one of those 9x9 tables, that would be very hard to find. And the problem would get worse as additional matrix types or operators were included.
The present version of newmat solves the problem by assigning attributes such as diagonal or band or upper triangular to each matrix type. Which attributes a matrix type has, is stored as bits in an integer. As an example, the DiagonalMatrix type has the bits corresponding to diagonal, symmetric and band equal to 1. By looking at the attributes of each of the operands of a binary operator, the program can work out the attributes of the result of the operation with simple bitwise operations. Hence it can deduce an appropriate type. The symmetric attribute is a minor problem because symmetric * symmetric does not yield symmetric unless both operands are diagonal. But otherwise very simple code can be used to deduce the attributes of the result of a binary operation.
Tables of the types resulting from the binary operators are output at the beginning of the test program.
Suppose you do something like
int* y = new int[100]; y += 200; // y points to something outside the array // y is never accessed
Then the standard says that the behaviour of the program is undefined even if y is never accessed. (You are allowed to calculate a pointer value one location beyond the end of the array). In practice, a program like this does not cause any problems with any compiler I have come across and no-one has reported any such problems to me.
However, this error is detected by Borland's Code Guard bound's checker and this makes it very difficult to use this to use Code Guard to detect other problems since the output is swamped by reports of this error.
Now consider
int* y = new int[100]; y += 200; // y points to something outside the array y -= 150; // y points to something inside the array // y is accessed
Again this is not strictly correct but does not seem to cause a problem. But it is much more doubtful than the previous example.
I removed most instances of the second version of the problem from Newmat09. Hopefully the remainder of these instances were removed from Newmat10. In addition, most instances of the first version of the problem have also been fixed.
There is one exception. The interface to the Numerical Recipes in C does still contain the second version of the problem. This is inevitable because of the way Numerical Recipes in C stores vectors and matrices. If you are running the test program with a bounds checking program, edit tmt.h to disable the testing of the NRIC interface.
The rule does does cause a problem for authors of matrix and multidimensional array packages. If we want to run down a column of a matrix we would like to do something like
// set values of column 1
Matrix A;
... set dimensions and put values in A
Real* a = A.Store(); // points to first element
int nr = A.Nrows(); // number of rows
int nc = A.Ncols(); // number of columns
while (nr--)
{
*a = something to put in first element of row
a += nc; // jump to next element of column
}
If the matrix has more than one column the last execution of a += nc; will run off the end of the space allocated to the matrix and we'll get a bounds error report.
Instead we have to use a program like
// set values of column 1
Matrix A;
... set dimensions and put values in A
Real* a = A.Store(); // points to first element
int nr = A.Nrows(); // number of rows
int nc = A.Ncols(); // number of columns
if (nr != 0)
{
for(;;)
{
*a = something to put in first element of row
if (!(--nr)) break;
a += nc; // jump to next element of column
}
}
which is more complicated and consequently introduces more chance of error.
The library now does have a moderately graceful exit from errors. One can use either the simulated exceptions or the compiler supported exceptions. When newmat08 was released (in 1995), compiler exception handling in the compilers I had access to was unreliable. I recommended you used my simulated exceptions. In 1997 compiler supported exceptions seemed to work on a variety of compilers - but not all compilers. This was still true in 2001. One compiler company was still having problems in 2003 (not sure about 2004). Try using the compiler supported exceptions if you have a recent compiler, but if you are getting strange crashes or errors try going back to my simulated exceptions.
The approach in the present library, attempting to simulate C++ exceptions, is not completely satisfactory, but seems a good interim solution for those who cannot use compiler supported exceptions. People who don't want exceptions in any shape or form, can set the option to exit the program if an exception is thrown.
The exception mechanism cannot clean-up objects explicitly created by new. This must be explicitly carried out by the package writer or the package user. I have not yet done this completely with the present package so occasionally a little garbage may be left behind after an exception. I don't think this is a big problem, but it is one that needs fixing.
The library does not support sparse matrices.
For sparse matrices there is going to be some kind of structure vector. It is going to have to be calculated for the results of expressions in much the same way that types are calculated. In addition, a whole new set of row and column operations would have to be written.
Sparse matrices are important for people solving large sets of differential equations as well as being important for statistical and operational research applications.
But there are packages being developed specifically for sparse matrices and these might present the best approach, at least where sparse matrices are the main interest.
The package does not yet support matrices with complex elements. There are at least two approaches to including these. One is to have matrices with complex elements.
This probably means making new versions of the basic row and column operations for Real*Complex, Complex*Complex, Complex*Real and similarly for + and -. This would be OK, except that if I also want to do this for sparse matrices, then when you put these together, the whole thing will get out of hand.
The alternative is to represent a Complex matrix by a pair of Real matrices. One probably needs another level of decoding expressions but I think it might still be simpler than the first approach. But there is going to be a problem with accessing elements and it does not seem possible to solve this in an entirely satisfactory way.
Complex matrices are used extensively by electrical engineers and physicists and really should be fully supported in a comprehensive package.
You can simulate most complex operations by representing Z = X + iY by
/ X Y \
\ -Y X /
Most matrix operations will simulate the corresponding complex operation, when applied to this matrix. But, of course, this matrix is essentially twice as big as you would need with a genuine complex matrix library.
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