//$$ newmat.txt            Documentation file


   Documentation for newmat03, an experimental matrix package in C++.
   ==================================================================


MATRIX PACKAGE                                           25 November, 1991

Copyright (C) 1991: R B Davies and DSIR

Permission is granted to use but not to sell.


Contents
========

General description
Is this the package you need?
Changes
Where you can get a copy of this package
Compiler performance
Example
Detailed documentation
   Customising
   Constructors
   Elements of matrices
   Matrix copy
   Unary operators
   Binary operators
   Combination of a matrix and scalar
   Scalar functions of matrices
   Submatrix operations
   Change dimensions
   Change type
   Multiple matrix solve
   Memory management
   Output
   Accessing matrices of unspecified type
   Cholesky decomposition
   Householder triangularisation
   Singular Value Decomposition
   Eigenvalues
   Sorting
   Fast Fourier Transform
   Interface to Numerical Recipes in C
List of files
Notes on the design of the package
   What this is package for
   What size of matrices?
   Allow matrix expressions?
   Which matrix types?
   What element types?
   Naming convention
   Row and Column index ranges
   Structure of matrix objects
   Data storage - one block or several
   Data storage - by row or by column or other
   Storage of symmetric matrices
   Element access - method and checking
   Use iterators?
   Memory management - reference counting or status variable?
   Evaluation of expressions - use two stage method?
   How to overcome an explosion in number of operations
   Using const
   A calculus of matrix types
   Error handling
   Band and sparse matrices
Problem report form


---------------------------------------------------------------------------


General description
===================

The package is intented 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)
    nricMatrix                   (variant of rectangular matrix)
    UpperTriangularMatrix
    LowerTriangularMatrix
    DiagonalMatrix
    SymmetricMatrix
    RowVector                    (derived from Matrix)
    ColumnVector                 (derived from Matrix).

Only one element type (float or double) is supported.

The package includes the operations *, +, -, inverse, transpose,
conversion between types, submatrix, determinant, Cholesky
decomposition, Householder triangularisation, 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 4 x 4 to about 90 x 90 (125 x
125 for triangular matrices). The upper limit is imposed by the maximum
number of elements that can be contained in a single array (8192 doubles
in some machines).

A two-stage approach to evaluating matrix expressions is used to improve
efficiency and reduce use of temporary storage.

The package is designed for version 2 of C++. It works with Turbo C++,
Borland C++, Glockenspiel C++ (2.00a) on a PC and AT&T C++ (2.0) and Gnu
C++ on a Sun. It works with some problems with Zortech C++ (version 2).


---------------------------------------------------------------------------


Is this the package you need?
=============================

Do you

1.   need matrix operators such as * and + defined as operators so you
     can write things like

        X  = A * (B + C);

2.   need a variety of types of matrices

3.   need only one element type (float or double)

4.   work with matrices in the range 4x4 to 90x90

5.   tolerate a large and complex package


Then maybe this is the right package for you. 

If you don't need (1) then there may be better options. Likewise if you
don't need (2) there may be better options. If you require "not (5)"
then this is not the package for you.


If you need (2) and "not (3)" and have some spare money, then maybe you
should look at M++ from Dyad or the Rogue Wave matrix package.


---------------------------------------------------------------------------


Changes
=======

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.


---------------------------------------------------------------------------


How to get a copy of this package
=================================

I am putting copies on Compuserve (Borland library, zip format),
SIMTEL20 (MsDos library, zip format), comp.sources.misc on Internet
(shar format), and on the MsDos program library at Victoria University,
Wellington.


---------------------------------------------------------------------------


Compiler performance
====================

I have tested this package on a number of compilers. Here are the
levels of success with this package. 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 MsDos
versions, I am using a 386/387sx computer running MsDos 5, except that
Turbo is on an old XT. The unix versions are on a Sun Sparc station.

A series of #defines at the beginning of "include.hxx" customises the
package for the compiler you are using. Turbo, Borland and Zortech are
recognised automatically, otherwise you have to set the appropriate
#define statement.

The compilers are looking a bit old now. I do intend to test the package
against newer versions as they become available.

Borland C++ 2.0: Recently, this has been my main development platform,
so naturally almost everything works with this compiler. The library
manager "tlib" fails but you can use "zorlib" from Zortech instead.
Sometimes Borland crashes during a compiler or mis-compiles. You just
have to reboot and continue the compile.

Turbo C++ (? version): Almost works OK. My rather elderly version does
show a problem. Probably not worth tracking down - buy a newer version.
Haven't tried the linker.

Zortech C++ 2.12: "const" doesn't work correctly with this compiler, so
the package skips all of the statements Zortech can't handle. If you are
using a later version of Zortech you could probably re-activate these
statements. Zortech leaves rubbish on the heap. I don't know whether
this is my programming error or a Zortech error. It works better when
one doesn't optimise but there still are problems. The nric routines
don't work. Zortech does not support IO manipulators.

Glockenspiel C++ (2.00a for MsDos loading into Microsoft C 5.1): I
haven't tested the latest version of my package with Glockenspiel. I had
to #define the matrix names to shorter names to avoid ambiguities and
had quite a bit of difficulty stopping the compiles from running out of
space and not exceeding Microsoft's block nesting limit. A couple of my
test statements produced statements too complex for Microsoft, but
basically the package worked. This was my original development platform
and I still use .cxx and .hxx as my file name extensions. However,
Glockenspiel is no longer competitive for MsDos and I am not updating my
copy of the compiler.

Sun AT&T C++ 2.00: This works fine. Except aggregates are not supported.

Gnu G++ 1.37.1: This mostly works. You don't seem to be able to use
expressions like Matrix(X*Y) in the middle of an expression and
(Matrix)(X*Y) is unreliable. Gnu does not support IO manipulators. Gnu
leaves rubbish on the heap. This is from output statements and not my
package and may not be an error. The previous version of the package did
not work under Gnu 1.37 or 1.39.


---------------------------------------------------------------------------

Example
=======

An example is given in  example.cxx .  This gives a simple linear
regression example using four different algorithms. The correct output
is given in example.txt. The program carries out a check that no memory
is left allocated on the heap when it terminates. The file  example.dep
contains a dependency list for compiling  example.cxx . You will need to
add the compile and link commands. example.dep lists all the files in
the package so you can adapt for other projects. Don't forget to remove
references to  newmat9.cxx  if you are using a compiler that does not
support the standard io manipulators.


---------------------------------------------------------------------------


Detailed Documentation
======================

Copyright (C) 1989,1990,1991: R B Davies and DSIR

Permission is granted to use but not to sell.

   --------------------------------------------------------------
  | Please understand that this is a test version; there may     |
  | still be bugs and errors. Use at your own risk. Neither I    |
  | nor DSIR take any 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@am.dsir.govt.nz

or

    Compuserve 72777,656

When reporting a bug please tell me which C++ compiler you are using (if
known), and what version. Also give me details of your computer (if
known). Tell me where you downloaded your version of my package from and
its version number (eg newmat03 or newmat04). (There may be very minor
differences between versions at different sites). Note any changes you
have made to my code. If at all possible give me a piece of code
illustrating the bug.

Please do report bugs to me.


The matrix inverse routine and the sort routines are adapted from
"Numerical Recipes in C" by Press, Flannery, Teukolsky, Vetterling,
published by the Cambridge University Press.

Other code is adapted from routines in "Handbook for Automatic
Computation, Vol II, Linear Algebra" by Wilkinson and Reinsch, published
by Springer Verlag. 


Customising
-----------

I use .hxx as the suffix of definition files and .cxx as the suffix of
C++ source files. This does not cause any problems with the compilers I
use except that Borland and Turbo need to be told to accept any suffix
as meaning a C++ file rather than a C file.

Use the large model when you are using a PC. Do not "outline" inline
functions.

Each file accessing the matrix package needs to have file newmat.hxx 
#included  at the beginning. Files using matrix applications (Cholesky
decomposition, Householder triangularisation) need newmatap.hxx instead
(or as well). If you need the output functions you will also need
newmatio.hxx. Glockenspiel also needs to have include.hxx #included before
newmat.hxx.

The file  include.hxx  sets the options for the compiler. If you are using
a compiler different from one I have worked with you may have to set up
a new section in  include.hxx  appropriate for your compiler.

Borland, Turbo and Zortech are recognised automatically. If you using
Glockenspiel on a PC, AT&T, or Gnu C++ activate the appropriate
statement at the beginning of include.hxx.

Activate the appropriate statement to make the element type float or
double.

The file (newmat9.cxx) containing the output routines can be used only
with libraries that support the AT&T input/output routines including
manipulators. It cannot be used with Zortech or Gnu.


Constructors
------------

To construct an m x n matrix, A, (m and n are integers) use

    Matrix A(m,n);

The UpperTriangularMatrix, LowerTriangularMatrix, SymmetricMatrix and
DiagonalMatrix types are symmetric. To construct an n x n matrix use,
for example

    UpperTriangularMatrix U(n);

Likewise the RowVector and ColumnVector types take just one argument in
their constructors:

    RowVector RV(n);

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-dimension statement.

You can also use a constructor to set a matrix equal to another matrix
or matrix expression.

    Matrix A = U;

    Matrix A = U * L;

Only conversions that don't lose information are supported - eg you
cannot convert an upper triangular matrix into a diagonal matrix using =.


Elements of matrices
--------------------

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 earlier 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.


Matrix copy
-----------

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.

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.

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; Matrix A(m,n);
    ......
    Matrix A(m,n); A = r;

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;

This construction cannot check that the numbers of elements match
correctly. This version of << can be used with submatrices on the left
hand side.


Unary operators
---------------

The package supports unary operations

    change sign of elements            -A
    transpose                          A.t()
    inverse (of square matrix A)       A.i()


Binary operations
-----------------

The package supports binary operations

    matrix addition                    A+B
    matrix subtraction                 A-B
    matrix multiplication              A*B
    equation solve (square matrix A)   A.i()*B

In the last case the inverse is not calculated.

Notes:

If you are doing repeated multiplication. For example A*B*C, use
brackets to force the order to minimize the number of operations. If C
is a column vector and A is not a vector, then it will usually reduce
the number of operations to use A*(B*C) .

The package does not recognise B*A.i() as an equation solve. It is
probably better to use (A.t().i()*B.t()).t() .


Combination of a matrix and scalar
----------------------------------

The following expression multiplies the elements of a matrix A by a
scalar f:  A * f; 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.

In each case the matrix must be the first term in the expression.
Expressions such  f + A  or  f * A  are not recognised.


Scalar functions of matrices
----------------------------
            
    int m = A.Nrows();                    // number of rows
    int n = A.Ncols();                    // number of columns
    real ssq = A.SumSquare();             // sum of squares of elements
    real sav = A.SumAbsoluteValue();      // sum of absolute values
    real mav = A.MaximumAbsoluteValue();  // maximum of absolute 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 t = A.Trace();                   // trace
    LogandSign ld = A.LogDeterminant();   // log of determinant
    BOOL z = A.IsZero();                  // test all elements zero
    MatrixType mt = A.Type();             // type of matrix
    real* s = Store();                    // pointer to array of elements
    int l = Storage();                    // length of array of elements

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.

A.IsZero() returns BOOL value TRUE if the matrix A has all elements
equal to 0.0.

MatrixType mt = A.Type() returns the type of a matrix. Use (char*)mt to
get a string  (UT, LT, Rect, Sym, Diag, RowV, ColV, Crout) showing the
type.

SumSquare(A), SumAbsoluteValue(A), MaximumAbsoluteValue(A), Trace(A),
LogDeterminant(A), Norm1(A), NormInfinity(A)  can be used in place of
A.SumSquare(), A.SumAbsoluteValue(), A.MaximumAbsoluteValue(),
A.Trace(), A.LogDeterminant(), A.Norm1(), A,NormInfinity().


Submatrix operations
--------------------

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. This may be used in any matrix expression or
on the left hand side of << or Inject. Inject does not check no
information loss in this case. You can also use the construction

    real c; .... A.SubMatrix(fr,lr,fc,lc) << c;

to set a submatrix equal to a constant.

The follwing 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).

If SubMatrix or its variant occurs on the right hand side of an = or <<
or within an expression its type is 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  << , 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.


Change dimensions
-----------------

The following operations change the dimensions of a matrix. The values
of the elements are lost.

    A.ReDimension(nrows,ncols);     // for type Matrix or nricMatrix
    A.ReDimension(n);               // for all other types


Change type
-----------

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.CopyToRow()
    A.CopyToColumn()
    A.CopyToDiagonal()
    A.CopyToMatrix(nrows,ncols)
    A.c()
    real(A)

The form .c() is used in matrix expressions when A is of a const
type. The expression real(A) is used to convert a 1 x 1 matrix to a
scalar.


Multiple matrix solve
---------------------

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 decompostion.


Memory management
-----------------

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)

Constant matrices cannot be used in matrix expressions so if you wish to
use A in an expression within this function use A.c() rather than 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;
       ......
       A.Release(); return A;
    }

or

    ReturnMatrix MakeAMatrix()
    {
       Matrix* m = new Matrix;
       ......
       m->ReleaseAndDelete(); return *m;
    }

Note that .c() cannot be applied to a matrix following application of
.Release() or ->ReleaseAndDelete() .

 --------------------------------------------------------------------- 
| Do not forget to make the function of type ReturnMatrix; otherwise  |
| incomprehensible run-time errors will occur with some compilers.    |
 --------------------------------------------------------------------- 

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
imporve 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.


Output
------

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 AT&T input/output
routines including manipulators.


Accessing matrices of unspecified type
--------------------------------------

Skip this section on your first reading.

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.cxx.


Cholesky decomposition
----------------------

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);


Householder triangularisation
-----------------------------

Start with matrix

       / X    0 \      s
       \ Y    0 /      t

         n    s

The Householder triangularisation post multiplies by an orthogonal
matrix Q such that the matrix becomes

       / 0    L \      s
       \ Z    M /      t

         n    s

where L is lower triangular. Note that X is the transpose of the matrix
sometimes considered in this context.

This is good for solving least squares problems: choose b (matrix or row
vector) to minimize the sum of the squares of the elements of

         Y - b*X

Then choose b = M * L.i();

Two routines are provided:

    HHDecompose(X, L);

replaces X by orthogonal columns and forms L.

    HHDecompose(X, Y, M);

uses X from the first routine, replaces Y by Z and forms M.


Singular Value Decomposition
----------------------------

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
diagonal matrix and V is an n x n orthogonal matrix.

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

The values of A are not changed unless A is also inserted as the third
argument.


Eigenvalues
-----------

An eigenvalue decomposition of a symmetric matrix A is a decomposition

    A  = V * D * V.t()

where V is an orthogonal matrix and D is a diagonal matrix.

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 is for workspace,
                                      //    S = A is OK
    Jacobi(A,D);                      // A symmetric
    Jacobi(A,D,S);                    // A, S symmetric is for workspace,
                                      //    S = A is OK
    Jacobi(A,D,V);                    // A symmetric

    EigenValues(A,D);                 // A symmetric
    EigenValues(A,D,S);               // A, S symmetric is for back
                                      //    transforming, S = A is OK
    EigenValues(A,D,V);               // A symmetric


Sorting
-------

To sort the values in a matrix or vector, A, (in general this operation
makes sense only for vectors and diagonal matrices) use

    SortAscending(A);

or

    SortDescending(A);


Fast Fourier Transform
----------------------

FFT(CV1, CV2, CV3, CV4);       // CV3=CV1 and CV4=CV2 is OK

where CV1, CV2, CV3, CV4 are column vectors. CV1 and CV2 are the real
and imaginary input vectors; CV3 and CV4 are the real and imaginary
output vectors. The lengths of CV1 and CV2 must be equal and should be
the product of numbers less than about 10 for fast execution.


Interface to 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 will also need to edit the function definitions to use
the version acceptable to your compiler. Then 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 replaced by a RowVector, ColumnVector or
DiagonalMatrix in the present package. Similarly any matrix with
subscripts starting from 1 can be replaced by an  nricMatrix  in the
present package. The class nricMatrix is derived from Matrix and can be
used in place of Matrix.

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,44,D,V,&nrot);          // jacobi from NRIC
   cout << D;                        // print eigenvalues

---------------------------------------------------------------------------


List of files
=============

README          readme file
NEWMAT   TXT    documentation file
NEWMAT   LIS    list of files
CLASS    TXT    list of classes

BOOLEAN  HXX    boolean class definition
CONTROLW HXX    control word definition file
INCLUDE  HXX    details of include files and options
NEWMAT   HXX    main matrix clss definition file
NEWMATAP HXX    applications definition file
NEWMATIO HXX    input/output definition file
NEWMATRC HXX    row/column functions definition files
NEWMATRM HXX    rectangular matrix access definition files
PRECISIO HXX    numerical precision constants

CHOLESKY CXX    Cholesky decomposition
EVALUE   CXX    eigenvalues and eigenvector calculation
FFT      CXX    fast Fourier transform
HHOLDER  CXX    Householder triangularisation
JACOBI   CXX    eigenvalues by the Jacobi method
NEWMAT1  CXX    type manipulation routines
NEWMAT2  CXX    row and column manipulation functions
NEWMAT3  CXX    row and column access functions
NEWMAT4  CXX    constructors, redimension, utilities
NEWMAT5  CXX    transpose, evaluate, matrix functions
NEWMAT6  CXX    operators, element access
NEWMAT7  CXX    invert, solve, binary operations
NEWMAT8  CXX    LU decomposition, scalar functions
NEWMAT9  CXX    output routines
NEWMATRM CXX    rectangular matrix access functions
SORT     CXX    sorting functions
SUBMAT   CXX    submatrix functions
SVD      CXX    singular value decomposition

EXAMPLE  CXX    example of use of package
EXAMPLE  TXT    output from example
EXAMPLE  DEP    dependency file for example

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Notes on the design of the package
==================================

Copyright (C) 1991: R B Davies and DSIR

Please treat this as an academic publication. You can use the ideas but
please acknowledge in any publication.


In this section, 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 section in order to use the
package.

I don't think it is obvious what is the best way of going about
structuring a matrix package. And 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.

But I do hope it will meet the immediate requirements of some people who
need to carry out matrix calculations.


What this is package for
------------------------

The package is used 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" package tends to be different from a general "array" package.

I see a matrix package as providing the following

1.   Evaluation of matrix expressions in a form familiar to
     scientists and engineers. For example  A = B * (C + D.t());
2.   Access to the elements of a matrix;
3.   Access to submatrices;
4.   Common elementary matrix functions such as determinant and trace;
5.   A platform for developing advanced matrix functions such as eigen-
     value analysis;
6.   Good efficiency and storage management;
7.   Graceful exit from errors (I don't provide this yet).

It may also provide

8.   A variety of types of elements (eg real and complex);
9.   A variety of types of matrices (eg rectangular and symmetric).

In contrast an array package should provide

1'.  Arrays can be copied with a single instruction; may have some
     arithmetic operations, say + - and scalar + - * /, it may provide
     matrix multiplication as a function;
2'.  High speed access to elements directly and with iterators;
3'.  Access to subarrays and rows (and columns?);
6'.  Good efficiency and storage management;
7'.  Graceful exit from errors;
8'.  A variety of types of elements (eg real and complex);
9'.  One, two and three dimensional arrays, at least, with starting
     points of the indices set by user; may have arrays with ragged
     rows.

It may be possible to amalgamate these two sets of requirements to some
extent. However my package is definitely oriented towards the first set.

Even within the bounds set by the first set of requirements 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 is 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.


What size of matrices?
----------------------

A matrix package may target small matrices (say 3 x 3), or medium sized
matrices, or very large matrices. A package targeting very small
matrices will seek to minimise administration. A package 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
package for very large matrices will need to pay special attention to
storage and numerical properties.

This package is designed for medium sized matrices. This means it is
worth introducing some optimisations, but I don't have to worry about
the 64K limit that some compilers impose.


Allow matrix expressions?
-------------------------

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.


Which matrix types?
-------------------

As well as the usual rectangular matrices, matrices occuring 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 package will need 25 versions of each of the binary
operators. In fact, we can evade this problem, but at the cost of some
complexity.


What element types?
-------------------

Ideally we would allow element types double, float, complex and int, at
least. It would be reasonably easy, using templates or equivalent, to
provide a package 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. 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 to float or double.

In retrospect, it would not be too hard to include matrices with complex
matrix type.

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
multistage Householder triangularisations.


Naming convention
-----------------

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.

The convention causes problems for Glockenspiel C++ on a PC feeding into
Microsoft C. The names Glockenspiel generates exceed the the 32
characters recognised by Microsoft C and ambiguities result. So it is
necessary to #define shorter names.

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() .


Row and Column index ranges
---------------------------

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.


Structure of matrix objects
---------------------------

Each matrix object contains the basic information such as the number of
rows and columns and a status variable plus a pointer to the data
array which is on the heap.


Data storage - one block or several
-----------------------------------

In this package 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 systems that
have a 64k byte (or other) limit on the size of arrays. The large arrays
may also cause problems for memory management in smaller machines.


Data storage - by row or by column or other
-------------------------------------------

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.

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.


Storage of symmetric matrices
-----------------------------

Symmetric matrices are stored as lower triangular matrices. The decision
was pretty arbitrary, but it does slightly simplify the Cholesky
decomposition program.


Element access - method and checking
------------------------------------

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 didn't even consider using the totally alien
notation A[i][j]. 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. This is how C works when you use A[i][j]. 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.


Use iterators?
--------------

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 for some row and column functions.


Memory management - reference counting or status variable?
----------------------------------------------------------

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 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 avoid one temporary being left over, so we would like this
temporary deleted as quickly as possible.

I provide the functionality for doing 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 delayed copying.
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. This saves the unnecessary copying in the
previous examples. However, it does not provide the additional
functionality of my approach.

It would be possible to have both delayed copy and tagging temporaries,
but this seemed an unnecessary complexity. In particular delayed copy
mechanisms seem to require two calls to the heap manager, using extra
time and making building a memory compacting mechanism more difficult.


Evaluation of expressions - use two stage method?
-------------------------------------------------

Consider the instruction

   X = B - X;

The 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 -= (which I haven't provided, yet)

   X -= B;

However this is an unnatural notation for non C users and it is much
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.

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.


How to overcome an explosion in number of operations
----------------------------------------------------

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.

An alternative approach of using iterators will be slower since the
iterators will involve a virtual function access at each step.

In fact, 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.


Using const
-----------

The memory management scheme introduces a problem when a matrix is
declared const. Because an operator may want to recycle the memory of
its operands these operands cannot be declared const. It isn't
reasonable for a temporary matrix to be declared const. However, I don't
know how to tell this to the C++ compiler. One possibility is to provide
alternative versions of the operators for operands declared const. But
then one gets the explosion in the number of operators.

My solution is to include versions of the initialisers for matrices
declared const. Otherwise, you need to use A.c() in place of A if A is
declared const and you wish to use it in an expression.


A calculus of matrix types
--------------------------

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 a temporary. 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.


Error handling
--------------

The package does not have graceful exit from errors. All errors are
treated as fatal. Originally I thought I would wait until exceptions
became available in C++. This now seems to have been delayed. In any
case I don't think exceptions will solve all the problems. Some clean up
of objects on the heap will often be required before one can exit via an
exception.

There are four categories of errors:

   Programming error - eg illegal conversion of a matrix, subscript out
   of bounds, matrix dimensions don't match;

   Illegal data error - eg Cholesky of a non-positive definite matrix;

   Out of space error - "new" returns a null pointer;

   Convergence failure - an iterative operation fails to converge.

For the first two of these, it might be sensible to terminate a program.
For the second two, one does want to return control to the user's
program in a convenient manner. I don't know a good way of doing this,
especially before exceptions are implemented.


Band and sparse matrices
------------------------

The package does not yet support band or sparse types. At present the
package assumes that the structure of a matrix is determined by its
class and dimensions. This is not sufficient for band and sparse
matrices.

For band matrices one also needs to know the upper and lower band
widths. For sparse matrices there is going to be some kind of structure
vector. These are 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
for sparse matrices. However the present ones will be fine for band
matrices.

Band and sparse matrices are important for people solving large
sets of differential equations. Sparse matrices are also important for
statistical and operational research applications.


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                   Matrix package problem report form
                   ----------------------------------

Version: ...............newmat03
Date of release: .......Nov 28th, 1991
Primary site: ..........comp.sources.misc
Downloaded from: .......
Your email address: ....
Today's date: ..........
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Describe the problem - attach examples if possible:









Email to  robert@am.dsir.govt.nz  or  Compuserve 72777,656 

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