!standard G.3 (01) 03-10-24 AI95-00296/06

!standard G.3.1 (01)

!standard G.3.2 (01)

!class amendment 02-06-07

!status work item 03-10-29

!status ARG Approved 10-0-0 03-10-03

!status work item 03-01-23

!status received 02-06-07

!priority Medium

!difficulty Medium

!subject Vector and matrix operations

!standard G.3.1 (01)

!standard G.3.2 (01)

!class amendment 02-06-07

!status work item 03-10-29

!status ARG Approved 10-0-0 03-10-03

!status work item 03-01-23

!status received 02-06-07

!priority Medium

!difficulty Medium

!subject Vector and matrix operations

!summary

The vector and matrix operations in ISO/IEC 13813 plus related operations
are added to Ada.Numerics in Annex G.

!problem

A number of secondary standards for Ada 83 were produced for the numerics area.
Most of the functionality of these standards was incorporated into Ada 95 (some
in the core language and some in the Numerics Annex) but two packages from
ISO/IEC 13813 were not. These are generic packages for the manipulation of real
and complex vectors and matrices.

The UK was asked to review the situation and to make a recommendation; they
recommended that if Ada were amended then consideration should be given to
including the packages within the Numerics annex.

The packages can be implemented entirely in Ada and thus present little burden
to implementors. Providing secondary standards has not proved satisfactory
because they are not sufficiently visible to the user community as a whole.

!proposal

It is proposed that two generic packages be added to the numerics annex. They
are Ada.Numerics.Generic_Real_Arrays and Ada.Numerics.Generic_Complex_Arrays.
They are included as a new subclause G.3 in order to avoid excessive
renumbering of other clauses.

In addition to the main facilities of 13813, these packages also include
subprograms for the solution of linear equations, matrix inversion,
determinants, and the determination of the eigenvalues and eigenvectors of
real symmetric matrices and Hermitian matrices.

!wording

Add a new clause G.3 as follows

G.3 Vector and Matrix Manipulation

Types and operations for the manipulation of real vectors and matrices are
provided in Generic_Real_Arrays, which is defined in G.3.1. Types and
operations for the manipulation of complex vectors and matrices are provided in
Generic_Complex_Arrays, which is defined in G.3.2. Both of these library units
are generic children of the predefined package Numerics (see A.5). Nongeneric
equivalents of these packages for each of the predefined floating point types
are also provided as children of Numerics.

G.3.1 Real Vectors and Matrices

The generic library package Numerics.Generic_Real_Arrays has the following
declaration:

-- Types

-- Subprograms for Real_Vector types

-- Real_Vector arithmetic operations

-- Real_Vector scaling operations

-- Other Real_Vector operations

-- Subprograms for Real_Matrix types

-- Real_Matrix arithmetic operations

-- Real_Matrix scaling operations

-- Real_Matrix inversion and related operations

-- Eigenvalues and vectors of a real symmetric matrix

-- Other Real_Matrix operations

-- implementation-defined

The library package Numerics.Real_Arrays is declared pure and defines the
same types and subprograms as Numerics.Generic_Real_Arrays, except that
the predefined type Float is systematically substituted for Real'Base
throughout. Nongeneric equivalents for each of the other predefined floating
point types are defined similarly, with the names Numerics.Short_Real_Arrays,
Numerics.Long_Real_Arrays, etc.

Two types are defined and exported by Ada.Numerics.Generic_Real_Arrays. The
composite type Real_Vector is provided to represent a vector with components of
type Real; it is defined as an unconstrained, one-dimensional array with an
index of type Integer. The composite type Real_Matrix is provided to represent
a matrix with components of type Real; it is defined as an unconstrained,
two-dimensional array with indices of type Integer.

The effect of the various functions is as described below. In most cases the
functions are described in terms of corresponding scalar operations of the type
Real; any exception raised by those operations is propagated by the array
operation. Moreover the accuracy of the result for each individual component is
as defined for the scalar operation unless stated otherwise.

In the case of those operations which are defined to involve an inner product,
Constraint_Error may be raised if an intermediate result is outside the range
of Real'Base even though the mathematical final result would not be.

function "+" (Right : Real_Vector) return Real_Vector;
function "-" (Right : Real_Vector) return Real_Vector;
function "abs" (Right : Real_Vector) return Real_Vector;

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Right. The index range of the result is
Right'Range.

function "+" (Left, Right : Real_Vector) return Real_Vector;
function "-" (Left, Right : Real_Vector) return Real_Vector;

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Left and the matching component of Right.
The index range of the result is Left'Range. The exception Constraint_Error
is raised if Left'Length is not equal to Right'Length.

function "*" (Left, Right : Real_Vector) return Real'Base;

This operation returns the inner product of Left and Right. The exception
Constraint_Error is raised if Left'Length is not equal to Right'Length. This
operation involves an inner product.

function "*" (Left : Real'Base; Right : Real_Vector) return Real_Vector;

This operation returns the result of multiplying each component of Right by the
scalar Left using the "*" operation of the type Real. The index range of the
result is Right'Range.

function "*" (Left : Real_Vector; Right : Real'Base) return Real_Vector;
function "/" (Left : Real_Vector; Right : Real'Base) return Real_Vector;

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Left and to the scalar Right. The index
range of the result is Left'Range.

This function returns a "unit vector" with Order components and a lower bound
of First. All components are set to 0.0 except for the Index component which is
set to 1.0. The exception Constraint_Error is raised if Index < First, Index >
First + Order - 1 or if First + Order - 1 > Integer'Last.

function "+" (Right : Real_Matrix) return Real_Matrix;
function "-" (Right : Real_Matrix) return Real_Matrix;
function "abs" (Right : Real_Matrix) return Real_Matrix;

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Right. The index ranges of the result are
those of Right.

This function returns the transpose of a matrix X. The first and second index
ranges of the result are X'Range(2) and X'Range(1) respectively.

function "+" (Left, Right : Real_Matrix) return Real_Matrix;
function "-" (Left, Right : Real_Matrix) return Real_Matrix;

Each operation returns the result of applying the corresponding operation of
type Real to each component of Left and the matching component of Right.
The index ranges of the result are those of Left. The exception
Constraint_Error is raised if Left'Length(1) is not equal to Right'Length(1) or
Left'Length(2) is not equal to Right'Length(2).

function "*" (Left, Right : Real_Matrix) return Real_Matrix;

This operation provides the standard mathematical operation for matrix
multiplication. The first and second index ranges of the result are
Left'Range(1) and Right'Range(2) respectively. The exception
Constraint_Error is raised if Left'Length(2) is not equal to Right'Length(1).
This operation involves inner products.

function "*" (Left, Right : Real_Vector) return Real_Matrix;

This operation returns the outer product of a (column) vector Left by a
(row) vector Right using the appropriate operation "*" of the type Real for
computing the individual components. The first and second index ranges of the
matrix result are Left'Range and Right'Range respectively.

function "*" (Left : Real_Vector; Right : Real_Matrix) return Real_Vector;

This operation provides the standard mathematical operation for multiplication
of a (row) vector Left by a matrix Right. The index range of the (row) vector
result is Right'Range(2). The exception Constraint_Error is raised if
Left'Length is not equal to Right'Length(1). This operation involves inner
products.

function "*" (Left : Real_Matrix; Right : Real_Vector) return Real_Vector;

This operation provides the standard mathematical operation for multiplication
of a matrix Left by a (column) vector Right. The index range of the (column)
vector result is Left'Range(1). The exception Constraint_Error is raised if
Left'Length(2) is not equal to Right'Length. This operation involves inner
products.

function "*" (Left : Real'Base; Right : Real_Matrix) return Real_Matrix;

This operation returns the result of multiplying each component of Right by the
scalar Left using the "*" operation of the type Real. The index ranges of the
matrix result are those of Right.

function "*" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix;
function "/" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix;

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Left and to the scalar Right. The index
ranges of the matrix result are those of Left.

This function returns a vector Y such that X is (nearly) equal to A * Y. This
is the standard mathematical operation for solving a single set of linear
equations. The index range of the result is X'Range. Constraint_Error is
raised if A'Length(1), A'Length(2) and X'Length are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.

This function returns a matrix Y such that X is (nearly) equal to A * Y. This
is the standard mathematical operation for solving several sets of linear
equations. The index ranges of the result are those of X. Constraint_Error
is raised if A'Length(1), A'Length(2) and X'Length(1) are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.

This function returns a matrix B such that A * B is (nearly) the unit matrix.
The index ranges of the result are those of A. Constraint_Error is raised if
A'Length(1) is not equal to A'Length(2). Constraint_Error is raised if
the matrix A is ill-conditioned.

This function returns the determinant of the matrix A. Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2).

This function returns the eigenvalues of the symmetric matrix A as a vector
sorted into order with the largest first. The exception Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2). The index range of the
result is A'Range(1). The exception Argument_Error is raised if the matrix A
is not symmetric.

This procedure computes both the eigenvalues and eigenvectors of the symmetric
matrix A. The out parameter Values is the same as that obtained by calling the
function Eigenvalues. The out parameter Vectors is a matrix whose columns are
the eigenvectors of the matrix A. The order of the columns corresponds to the
order of the eigenvalues. The eigenvectors are normalized and mutually
orthogonal (they are orthonormal), including when there are repeated
eigenvalues. The exception Constraint_Error is raised if A'Length(1) is not
equal to A'Length(2). The index ranges of the parameter Vectors are those of A.
The exception Argument_Error is raised if the matrix A is not symmetric.

This function returns a square "unit matrix" with Order**2 components and
lower bounds of First_1 and First_2 (for the first and second index ranges
respectively). All components are set to 0.0 except for the main diagonal,
whose components are set to 1.0. The exception Constraint_Error is raised if
First_1 + Order - 1 > Integer'Last or First_2 + Order - 1 > Integer'Last.

Implementation Requirements

Accuracy requirements for the subprograms Solve, Inverse, Determinant,
Eigenvalues and Eigensystem are implementation defined.

For operations not involving an inner product, the accuracy
requirements are those of the corresponding operations of the type Real in
both the strict mode and the relaxed mode (see G.2).

For operations involving an inner product, no requirements are specified in
the relaxed mode. In the strict mode the modulus of the absolute error of
the inner product X*Y shall not exceed g*abs(X)*abs(Y) where g is defined as

g = X'Length * Machine_Radix**(1-Machine_Mantissa)

Implementations shall document any techniques used to reduce cancellation
errors such as extended precision arithmetic.

AARM Note

The above accuracy requirement is met by the canonical implementation of the
inner product by multiplication and addition using the corresponding
operations of type Real'Base and performing the cumulative addition using
ascending indices. Note however, that some hardware provides special
operations for the computation of the inner product and although these may be
fast they may not meet the accuracy requirement specified. See Accuracy and
Stability of Numerical Algorithms By N J Higham (ISBN 0-89871-355-2),
Section 3.1.

Implementation Permissions

The nongeneric equivalent packages may, but need not, be actual
instantiations of the generic package for the appropriate predefined type.

Implementation Advice

Implementations should implement the Solve and Inverse functions using
established techniques such as LU decomposition with row interchanges followed
by back and forward substitution. Implementations are recommended to refine the
result by performing an iteration on the residuals; if this is done then it
should be documented.

It is not the intention that any special provision should be made to
determine whether a matrix is ill-conditioned or not. The naturally occurring
overflow (including division by zero) which will result from executing these
functions with an ill-conditioned matrix and thus raise Constraint_Error is
sufficient.

The test that a matrix is symmetric may be performed by using the equality
operator to compare the relevant components.

G.3.2 Complex Vectors and Matrices

The generic library package Numerics.Generic_Complex_Arrays has the following
declaration:

-- Types

-- Subprograms for Complex_Vector types

-- Complex_Vector selection, conversion and composition operations

-- Complex_Vector arithmetic operations

-- Mixed Real_Vector and Complex_Vector arithmetic operations

-- Complex_Vector scaling operations

-- Other Complex_Vector operations

-- Subprograms for Complex_Matrix types

-- Complex_Matrix selection, conversion and composition operations

-- Complex_Matrix arithmetic operations

-- Mixed Real_Matrix and Complex_Matrix arithmetic operations

-- Complex_Matrix scaling operations

-- Complex_Matrix inversion and related operations

-- Eigenvalues and vectors of a Hermitian matrix

-- Other Complex_Matrix operations

The library package Numerics.Complex_Arrays is declared pure and defines
the same types and subprograms as Numerics.Generic_Complex_Arrays, except
that the predefined type Float is systematically substituted for Real'Base,
and the Real_Vector and Real_Matrix types exported by Numerics.Real_Arrays
are systematically substituted for Real_Vector and Real_Matrix, and the
Complex type exported by Numerics.Complex_Types is systematically
substituted for Complex, throughout. Nongeneric equivalents for each of
the other predefined floating point types are defined similarly, with the
names Numerics.Short_Complex_Arrays, Numerics.Long_Complex_Arrays, etc.

Two types are defined and exported by Ada.Numerics.Generic_Complex_Arrays.
The composite type Complex_Vector is provided to represent a vector with
components of type Complex; it is defined as an unconstrained
one-dimensional array with an index of type Integer. The composite type
Complex_Matrix is provided to represent a matrix with components of type
Complex; it is defined as an unconstrained, two-dimensional array with
indices of type Integer.

The effect of the various subprograms is as described below. In many cases
they are described in terms of corresponding scalar operations in
Numerics.Generic_Complex_Types. Any exception raised by those operations is
propagated by the array subprogram. Moreover any constraints on the parameters
and the accuracy of the result for each individual component is as defined for
the scalar operation.

In the case of those operations which are defined to involve an inner product,
Constraint_Error may be raised if an intermediate result has a component
outside the range of Real'Base even though the final mathematical result
would not.

Each function returns a vector of the specified cartesian components of X.
The index range of the result is X'Range.

Each procedure replaces the specified (cartesian) component of each of
the components of X by the value of the matching component of Re or Im;
the other (cartesian) component of each of the components is unchanged.
The exception Constraint_Error is raised if X'Length is not equal to
Re'Length or Im'Length.

Each function constructs a vector of Complex results (in cartesian
representation) formed from given vectors of cartesian components; when
only the real components are given, imaginary components of zero are assumed.
The index range of the result is Re'Range. The exception Constraint_Error is
raised if Re'Length is not equal to Im'Length.

Each function calculates and returns a vector of the specified polar
components of X or Right using the corresponding function in
Numerics.Generic_Complex_Types. The index range of the result is X'Range or
Right'Range.

Each function constructs a vector of Complex results (in cartesian
representation) formed from given vectors of polar components using the
corresponding function in Numerics.Generic_Complex_Types on matching
components of Modulus and Argument. The index range of the result is
Modulus'Range. The exception Constraint_Error is raised if
Modulus'Length is not equal to Argument'Length.

function "+" (Right : Complex_Vector) return Complex_Vector;
function "-" (Right : Complex_Vector) return Complex_Vector;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Right. The index
range of the result is Right'Range.

This function returns the result of applying the appropriate function
Conjugate in Numerics.Generic_Complex_Types to each component of X. The
index range of the result is X'Range.

function "+" (Left, Right : Complex_Vector) return Complex_Vector;
function "-" (Left, Right : Complex_Vector) return Complex_Vector;

Each operation returns the result of applying the corresponding operation
in Numerics.Generic_Complex_Types to each component of Left and the
matching component of Right. The index range of the result is Left'Range.
The exception Constraint_Error is raised if Left'Length is not equal to
Right'Length.

function "*" (Left, Right : Complex_Vector) return Complex;

This operation returns the inner product of Left and Right. The exception
Constraint_Error is raised if Left'Length is not equal to Right'Length.
This operation involves an inner product.

function "+" (Left : Real_Vector;

Right : Complex_Vector) **return** Complex_Vector;

function "+" (Left : Complex_Vector;

Right : Real_Vector) **return** Complex_Vector;

function "-" (Left : Real_Vector;

Right : Complex_Vector) **return** Complex_Vector;

function "-" (Left : Complex_Vector;

Right : Real_Vector) **return** Complex_Vector;

Each operation returns the result of applying the corresponding operation
in Numerics.Generic_Complex_Types to each component of Left and the
matching component of Right. The index range of the result is Left'Range.
The exception Constraint_Error is raised if Left'Length is not equal to
Right'Length.

function "*" (Left : Real_Vector; Right : Complex_Vector) return Complex;
function "*" (Left : Complex_Vector; Right : Real_Vector) return Complex;

Each operation returns the inner product of Left and Right. The exception
Constraint_Error is raised if Left'Length is not equal to Right'Length.
These operations involve an inner product.

function "*" (Left : Complex; Right : Complex_Vector) return Complex_Vector;

This operation returns the result of multiplying each component of Right by
the complex number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index range of the result is Right'Range.

function "*" (Left : Complex_Vector; Right : Complex) return Complex_Vector;
function "/" (Left : Complex_Vector; Right : Complex) return Complex_Vector;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the vector Left and the
complex number Right. The index range of the result is Left'Range.

function "*" (Left : Real'Base; Right : Complex_Vector) return Complex_Vector;

This operation returns the result of multiplying each component of Right by
the real number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index range of the result is Right'Range.

function "*" (Left : Complex_Vector; Right : Real'Base) return Complex_Vector;
function "/" (Left : Complex_Vector; Right : Real'Base) return Complex_Vector;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the vector Left and the
real number Right. The index range of the result is Left'Range.

This function returns a "unit vector" with Order components and a lower
bound of First. All components are set to (0.0,0.0) except for the Index
component which is set to (1.0,0.0). The exception Constraint_Error is
raised if Index < First, Index > First + Order - 1, or
if First + Order - 1 > Integer'Last.

Each function returns a matrix of the specified cartesian components of X.
The index ranges of the result are those of X.

Each procedure replaces the specified (cartesian) component of each of the
components of X by the value of the matching component of Re or Im; the other
(cartesian) component of each of the components is unchanged. The exception
Constraint_Error is raised if X'Length(1) is not equal to Re'Length(1) or
Im'Length(1) or if X'Length(2) is not equal to Re'Length(2) or Im'Length(2).

Each function constructs a matrix of Complex results (in cartesian
representation) formed from given matrices of cartesian components; when
only the real components are given, imaginary components of zero are
assumed. The index ranges of the result are those of Re. The exception
Constraint_Error is raised if Re'Length(1) is not equal to Im'Length(1) or
Re'Length(2) is not equal to Im'Length(2).

Each function calculates and returns a matrix of the specified polar
components of X or Right using the corresponding function in
Numerics.Generic_Complex_Types. The index ranges of the result are those of X
or Right.

Each function constructs a matrix of Complex results (in cartesian
representation) formed from given matrices of polar components using
the corresponding function in Numerics.Generic_Complex_Types on matching
components of Modulus and Argument. The index ranges of the result are those
of Modulus. The exception Constraint_Error is raised if Modulus'Length(1) is
not equal to Argument'Length(1) or Modulus'Length(2) is not equal to
Argument'Length(2).

function "+" (Right : Complex_Matrix) return Complex_Matrix;
function "-" (Right : Complex_Matrix) return Complex_Matrix;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Right. The index
ranges of the result are those of Right.

This function returns the result of applying the appropriate function Conjugate
in Numerics.Generic_Complex_Types to each component of X. The index ranges of
the result are those of X.

This function returns the transpose of a matrix X. The first and second index
ranges of the result are X'Range(2) and X'Range(1) respectively.

function "+" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "-" (Left, Right : Complex_Matrix) return Complex_Matrix;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Left and the matching
component of Right. The index ranges of the result are those of
Left. The exception Constraint_Error is raised if Left'Length(1) is not equal to
Right'Length(1) or Left'Length(2) is not equal to Right'Length(2).

function "*" (Left, Right : Complex_Matrix) return Complex_Matrix;

This operation provides the standard mathematical operation for matrix
multiplication. The first and second index ranges of the result are
Left'Range(1) and Right'Range(2) respectively. The exception
Constraint_Error is raised if Left'Length(2) is not equal to Right'Length(1).
This operation involves inner products.

function "*" (Left, Right : Complex_Vector) return Complex_Matrix;

This operation returns the outer product of a (column) vector Left by a (row)
vector Right using the appropriate operation "*" in
Numerics.Generic_Complex_Types for computing the individual components.
The first and second index ranges of the matrix result are Left'Range and
Right'Range respectively.

function "*" (Left : Complex_Vector;

Right : Complex_Matrix) **return** Complex_Vector;

This operation provides the standard mathematical operation for multiplication
of a (row) vector Left by a matrix Right. The index range of the (row) vector
result is Right'Range(2). The exception Constraint_Error is raised if
Left'Length is not equal to Right'Length(1). This operation involves inner
products.

function "*" (Left : Complex_Matrix;

Right : Complex_Vector) **return** Complex_Vector;

This operation provides the standard mathematical operation for multiplication
of a matrix Left by a (column) vector Right. The index range of the (column)
vector result is Left'Range(1). The exception Constraint_Error is raised if
Left'Length(2) is not equal to Right'Length. This operation involves inner
products.

function "+" (Left : Real_Matrix;

Right : Complex_Matrix) **return** Complex_Matrix;

function "+" (Left : Complex_Matrix;

Right : Real_Matrix) **return** Complex_Matrix;

function "-" (Left : Real_Matrix;

Right : Complex_Matrix) **return** Complex_Matrix;

function "-" (Left : Complex_Matrix;

Right : Real_Matrix) **return** Complex_Matrix;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Left and the matching
component of Right. The index ranges of the result are those of Left. The
exception Constraint_Error is raised if Left'Length(1) is not equal to
Right'Length(1) or Left'Length(2) is not equal to Right'Length(2).

function "*" (Left : Real_Matrix;

Right : Complex_Matrix) **return** Complex_Matrix;

function "*" (Left : Complex_Matrix;

Right : Real_Matrix) **return** Complex_Matrix;

Each operation provides the standard mathematical operation for matrix
multiplication. The first and second index ranges of the result are
Left'Range(1) and Right'Range(2) respectively. The exception
Constraint_Error is raised if Left'Length(2) is not equal to Right'Length(1).
These operations involve inner products.

function "*" (Left : Real_Vector;

Right : Complex_Vector) **return** Complex_Matrix;

function "*" (Left : Complex_Vector;

Right : Real_Vector) **return** Complex_Matrix;

Each operation returns the outer product of a (column) vector Left by a (row)
vector Right using the appropriate operation "*" in
Numerics.Generic_Complex_Types for computing the individual components.
The first and second index ranges of the matrix result are Left'Range and
Right'Range respectively.

function "*" (Left : Real_Vector;

Right : Complex_Matrix) **return** Complex_Vector;

function "*" (Left : Complex_Vector;

Right : Real_Matrix) **return** Complex_Vector;

Each operation provides the standard mathematical operation for multiplication
of a (row) vector Left by a matrix Right. The index range of the (row) vector
result is Right'Range(2). The exception Constraint_Error is raised if
Left'Length is not equal to Right'Length(1). These operations involve inner
products.

function "*" (Left : Real_Matrix;

Right : Complex_Vector) **return** Complex_Vector;

function "*" (Left : Complex_Matrix;

Right : Real_Vector) **return** Complex_Vector;

Each operation provides the standard mathematical operation for multiplication
of a matrix Left by a (column) vector Right. The index range of the (column)
vector result is Left'Range(1). The exception Constraint_Error is raised if
Left'Length(2) is not equal to Right'Length. These operations involve inner
products.

function "*" (Left : Complex; Right : Complex_Matrix) return Complex_Matrix;

This operation returns the result of multiplying each component of Right by
the complex number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index ranges of the result are those of
Right.

function "*" (Left : Complex_Matrix; Right : Complex) return Complex_Matrix;
function "/" (Left : Complex_Matrix; Right : Complex) return Complex_Matrix;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the matrix Left and the
complex number Right. The index ranges of the result are those of Left.

function "*" (Left : Real'Base; Right : Complex_Matrix) return Complex_Matrix;

This operation returns the result of multiplying each component of Right by
the real number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index ranges of the result are those of
Right.

function "*" (Left : Complex_Matrix; Right : Real'Base) return Complex_Matrix;
function "/" (Left : Complex_Matrix; Right : Real'Base) return Complex_Matrix;

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the matrix Left and the
scalar Right. The index ranges of the result are those of Left.

This function returns a vector Y such that X is (nearly) equal to A * Y. This
is the standard mathematical operation for solving a single set of linear
equations. The index range of the result is X'Range. Constraint_Error is
raised if A'Length(1), A'Length(2) and X'Length are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.

This function returns a matrix Y such that X is (nearly) equal to A * Y. This
is the standard mathematical operation for solving several sets of linear
equations. The index ranges of the result are those of X. Constraint_Error
is raised if A'Length(1), A'Length(2) and X'Length(1) are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.

This function returns a matrix B such that A * B is (nearly) the unit matrix.
The index ranges of the result are those of A. Constraint_Error is raised if
A'Length(1) is not equal to A'Length(2). Constraint_Error is raised if
the matrix A is ill-conditioned.

This function returns the determinant of the matrix A. Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2).

This function returns the eigenvalues of the Hermitian matrix A as a vector
sorted into order with the largest first. The exception Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2). The index range of the
result is A'Range(1). The exception Argument_Error is raised if the matrix A
is not Hermitian.

This procedure computes both the eigenvalues and eigenvectors of the Hermitian
matrix A. The out parameter Values is the same as that obtained by calling the
function Eigenvalues. The out parameter Vectors is a matrix whose columns are
the eigenvectors of the matrix A. The order of the columns corresponds to the
order of the eigenvalues. The eigenvectors are mutually orthonormal,
including when there are repeated eigenvalues. The exception Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2). The index ranges of the
parameter Vectors are those of A. The exception Argument_Error is raised if the
matrix A is not Hermitian.

This function returns a square "unit matrix" with Order**2 components and
lower bounds of First_1 and First_2 (for the first and second index ranges
respectively). All components are set to (0.0,0.0) except for the main diagonal,
whose components are set to (1.0,0.0). The exception Constraint_Error is raised
if First_1 + Order - 1 > Integer'Last or First_2 + Order - 1 > Integer'Last.

Implementation Requirements

Accuracy requirements for the subprograms Solve, Inverse, Determinant,
Eigenvalues and Eigensystem are implementation defined.

For operations not involving an inner product, the accuracy requirements are
those of the corresponding operations of the type Real'Base and Complex in both
the strict mode and the relaxed mode (see G.2).

For operations involving an inner product, no requirements are specified in
the relaxed mode. In the strict mode the modulus of the absolute error of the
inner product X*Y shall not exceed g*abs(X)*abs(Y) where g is defined as

g = sqrt(2.0) * X'Length * Machine_Radix**(1-Machine_Mantissa) for mixed

complex and real operands

g = sqrt(8.0) * X'Length * Machine_Radix**(1-Machine_Mantissa) for two

complex operands

Implementations shall document any techniques used to reduce cancellation
errors such as extended precision arithmetic.

AARM Note

The above accuracy requirement is met by the canonical implementation of the
inner product by multiplication and addition using the corresponding
operations of type Complex and performing the cumulative addition using
ascending indices. Note however, that some hardware provides special
operations for the computation of the inner product and although these may be
fast they may not meet the accuracy requirement specified. See Accuracy and
Stability of Numerical Algorithms by N J Higham (ISBN 0-89871-355-2),
Sections 3.1 and 3.6.

Implementation Permissions

The nongeneric equivalent packages may, but need not, be actual
instantiations of the generic package for the appropriate predefined type.

Although many operations are defined in terms of operations from
Numerics.Generic_Complex_Types, they need not be implemented by calling those
operations provided that the effect is the same.

Implementation Advice

Implementations should implement the Solve and Inverse functions using
established techniques. Implementations are recommended to refine the result by
performing an iteration on the residuals; if this is done then it should be
documented.

It is not the intention that any special provision should be made to
determine whether a matrix is ill-conditioned or not. The naturally occurring
overflow (including division by zero) which will result from executing these
functions with an ill-conditioned matrix and thus raise Constraint_Error is
sufficient.

The test that a matrix is Hermitian may use the equality operator to compare
the real components and negation followed by equality to compare the imaginary
components (see G.2.1).

Implementations should not perform operations on mixed complex and real operands
by first converting the real operand to complex. See G.1.1(56, 57).

!example

The packages are self-explanatory and so no example is provided.

!discussion

Section G.1.1 of the Rationale for Ada 95 says

"A decision was made to abbreviate the Ada 95 packages by omitting the vector
and matrix types and operations. One reason was that such types and operations
were largely self-evident, so that little real help would be provided by
defining them in the language. Another reason was that a future version of Ada
might add enhancements for array manipulation and so it would be inappropriate
to lock in such operations prematurely."

It is now clear that such enhancements will not be added so the second
justification for omitting the facilities of 13813 disappears. In order to
overcome the objection that everything is self-evident we have taken the
approach that we should add some basic facilities that are commonly required,
not completely trivial to implement but on the other hand are mathematically
well understood.

The overall goal is thus twofold

* to provide commonly required facilities for the user who is not a numerical

professional,

* to provide a baseline of types and operations that form a firm foundation

for binding to more general facilities such as the well-known BLAS (Basic
Linear Algebra Subprograms, see www.netlib.org/blas).

The packages closely follow those in 13813. However, the discussion has been
considerably simplified by assuming the properties of the corresponding scalar
operations such as those in Numerics.Complex_Types. This removes a lot of
explicit mention of raising exceptions and accuracy. Also remarks that these
are standard mathematical operations have been deleted when the meaning is
given by other words in the description.

The component by component operations of * / and ** on vectors have been
deleted on the grounds that they are not useful. (They might be useful for
manipulating arrays in general but we are concerned with arrays used as vectors
for linear algebra.)

Operations for vector products were considered but not added. This is because,
as usually defined, they only apply in three-dimensional space.

It is hoped that there is not too much confusion between component when
applied to the parts of a complex number and component when applied to a part
of an array. 13813 uses element for the latter but the proper Ada term for an
element of an array or field of a record is of course component.

Observe that the index range of the various arrays is Integer (rather than
Natural or Positive). This permits negative indices and in particular arrays
with symmetric index ranges about zero.

The function Identity_Matrix of 13813 has been changed to Unit_Matrix on the
grounds that the prime concern is with linear algebra and not group theory.

The accuracy of most simple operations follows from the underlying operations
on scalar components. In the case of inner product there is the potential for
special operations to improve the speed and/or accuracy. We have specified
reasonable requirements in the exact mode, which are met by the canonical
implementation using a loop statement. This is because on some
hardware, built-in instructions which are fast actually lose accuracy. Note
that the Fortran language standard recognizes the existence of inner product
as a special case but imposes no accuracy requirements at all.

These requirements for inner product are based on the analysis in Chapter 3
of Accuracy and Stability of Numerical Algorithms by N J Higham. A factor of
nearly 2 has been added as a precaution. In the complex case the factor is
increased by a factor of sqrt(2) in the mixed complex and real case to take
account of the hypotenuse effect and by a further factor of 2 in the pure
complex case since the number of terms is effectively doubled.

Functions have been added to Numerics.Generic_Real_Arrays for matrix inversion
and related operations. On the grounds that it would be helpful to have simple
algorithms available to users rather than none at all, no accuracy
requirements are specified. Instead the accuracy is stated to be
implementation-defined which means that the implementation must document it.

The names chosen are Solve and Inverse. The former is overloaded, one version
solves a single set of linear equations; the second solves multiple sets. Note
that Inverse is not strictly necessary because the effect can be obtained by
using Solve and a Unit_Matrix thus

I := Unit_Matrix(A'Length); B := Solve(A, I); -- same as B := Inverse (A);

A common technique when solving sets of linear equations is to refine the result
by an iteration on the residuals. Thus to solve the set of equations Ax = y we
first perform

X := Solve(A, Y);

we then compute the error D by multiplying back thus

D := Y - A*X;

and then compute the refinement to X

DX := Solve(A, D);

and then correct X thus

X := X + DX;

Implementations are recommended to do this automatically; it requires little
extra computation if LU decomposition is used internally. If they do such
refinement then it should be documented.

A function for computing the determinant has been added since it is of some
help in deciding whether an array is ill-conditioned and therefore the results
of inversion might be suspect. Determinants are also useful in some statistical
calculations. The evaluation of determinants is very liable to overflow and many
such routines return a scaling power of 10 in order to keep the basic result
within range. For simplicity, it was decided not to do this since it is less
likely with matrices of low order; the user can of course scale the components
of the matrix if necessary.

Similar functions have also been added for complex arrays. However, it was not
deemed necessary to provide for mixed real and complex operands for Solve.

In addition, subprograms have been added for the computation of eigenvalues
and vectors of real symmetric matrices and Hermitian matrices. The subprograms
are Eigenvalues and Eigensystem. There is no separate subprogram for
eigenvectors since it is unlikely that these would be required without the
eigenvalues.

The most common application is with real symmetric matrices and these have real
eigenvalues and vectors. Applications include: moments of inertia, elasticity,
confidence regions and so on.

A slight problem arises in deciding who should check that a matrix is symmetric,
the user or the package. Computations of, for example, a covariance matrix will
result in a matrix that ought to be exactly symmetric but small errors might be
introduced which mean that it is not exactly symmetric. The onus could be
placed on the user to ensure that the matrix is exactly symmetric or
alternatively some tolerance could be passed to the eigen subprograms so that
they can perform a reasonable check. Passing a tolerance level adds an
irritating parameter, raises the issue of how the tolerance should be expressed
and gives the user the problem of what tolerance to request. Moreover, the
algorithms still have to decide which actual values should be used for those
pairs of components that are not exactly equal.

Accordingly, we have placed the onus on the user to ensure that the matrix is
exactly symmetric. This could be done for example by taking the mean of the
matrix and its transpose. The test in the subprograms can then test for exact
equality which is guaranteed to deliver the correct answer. If the test fails
then Argument_Error is raised. This exception, declared in Ada.Numerics, is
the normal exception for numeric operations when the argument is out of range.
Note that errors such as when bounds of arrays do not match raise
Constraint_Error by analogy with built-in array operations.

A third approach considered was for the user to supply a parameter indicating
which half of the matrix should be used to define it (the upper or lower
triangle). This avoids the need for any testing but it was considered bad
practice to be able to pass junk in the other half of the matrix.

The complex equivalent of real symmetric matrices are Hermitian matrices.
Hermitian matrices are such that their transpose (that is with rows and columns
interchanged) equals their complex conjugate (that is with the sign of imaginary
parts reversed). Hermitian matrices also have real eigenvalues.
Applications include Quantum Mechanics.

Again we have placed the onus on the user to ensure that the matrix is
Hermitian. The check in the package can then use strict equality for the real
parts and negation followed by equality for the imaginary parts.

We considered providing subprograms for the determination of eigenvalues and
eigenvectors of general real and complex matrices. Such matrices can have
complex eigenvalues and therefore provision for these would have to be in the
complex package. However, there are mathematical difficulties with these general
cases which are in strong contrast to the real symmetric and Hermitian matrices.
Thus, Numerical Recipes by Press, Flannery, Teukolsky and Vetterling says
regarding the real case:

"The algorithms for symmetric matrices ... are highly satisfactory in practice.
By contrast, it is impossible to design equally satisfactory algorithms for the
nonsymmetric case. There are two reasons for this. First, the eigenvalues of a
nonsymmetric matrix can be very sensitive to small changes in the matrix
elements. Second, the matrix itself can be defective so that there is no
complete set of eigenvectors. We emphasize that these difficulties are intrinsic
properties of certain nonsymmetric matrices, and no numerical procedure can cure
them."

Similar remarks apply to complex matrices where Hermitian matrices are well-
behaved but non-Hermitian matrices can be troublesome.

In view of these computational difficulties and the fact that requiring the
eigensystem of general matrices is uncommon, we decided not to provide such
facilities.

Consideration was also given to the inclusion of explicit facilities for LU
decomposition (as provided for example in the BLAS). LU decomposition is a
common first step for many operations. Thus making it available to the user
permits more rapid computation when several operations such as solving
equations and determinant evaluation are to be performed using
the same matrix. However, modern hardware is so fast that this would only
seem to be necessary for very large sets of equations and these are not the
target of these simple facilities. Moreover, adding explicit LU decomposition
introduces complexity for the user.

Consideration was also given to a fuller implementation of the BLAS. However,
this seems out of place in a language standard since it would be extremely long
and specialized. Such a fuller interface to the BLAS could be provided in
additional child packages. The goal here has been to provide convenient access
to simple type declarations and the more commonly required operations that are
not trivial for the user to program. These operations can of course be
implemented as a binding to an implementation of part of the BLAS.

!corrigendum G.3(01)

Insert new clause:

Types and operations for the manipulation of real vectors and matrices are
provided in Generic_Real_Arrays, which is defined in G.3.1. Types and
operations for the manipulation of complex vectors and matrices are provided in
Generic_Complex_Arrays, which is defined in G.3.2. Both of these library units
are generic children of the predefined package Numerics (see A.5). Nongeneric
equivalents of these packages for each of the predefined floating point types
are also provided as children of Numerics.

!corrigendum G.3.1(01)

Insert new clause:

The generic library package Numerics.Generic_Real_Arrays has the following
declaration:

--

--

--

--

--

--

--

--

--

--

--

--

The library package Numerics.Real_Arrays is declared pure and defines the
same types and subprograms as Numerics.Generic_Real_Arrays, except that
the predefined type Float is systematically substituted for Real'Base
throughout. Nongeneric equivalents for each of the other predefined floating
point types are defined similarly, with the names Numerics.Short_Real_Arrays,
Numerics.Long_Real_Arrays, etc.

Two types are defined and exported by Ada.Numerics.Generic_Real_Arrays. The
composite type Real_Vector is provided to represent a vector with components of
type Real; it is defined as an unconstrained, one-dimensional array with an
index of type Integer. The composite type Real_Matrix is provided to represent
a matrix with components of type Real; it is defined as an unconstrained,
two-dimensional array with indices of type Integer.

The effect of the various functions is as described below. In most cases the
functions are described in terms of corresponding scalar operations of the type
Real; any exception raised by those operations is propagated by the array
operation. Moreover the accuracy of the result for each individual component is
as defined for the scalar operation unless stated otherwise.

In the case of those operations which are defined to involve an inner product,
Constraint_Error may be raised if an intermediate result is outside the range
of Real'Base even though the mathematical final result would not be.

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Right. The index range of the result is
Right'Range.

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Left and the matching component of Right.
The index range of the result is Left'Range. The exception Constraint_Error
is raised if Left'Length is not equal to Right'Length.

This operation returns the inner product of Left and Right. The exception
Constraint_Error is raised if Left'Length is not equal to Right'Length. This
operation involves an inner product.

This operation returns the result of multiplying each component of Right by the
scalar Left using the "*" operation of the type Real. The index range of the
result is Right'Range.

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Left and to the scalar Right. The index
range of the result is Left'Range.

This function returns a "unit vector" with Order components and a lower bound
of First. All components are set to 0.0 except for the Index component which is
set to 1.0. The exception Constraint_Error is raised if Index < First, Index >
First + Order - 1 or if First + Order - 1 > Integer'Last.

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Right. The index ranges of the result are
those of Right.

This function returns the transpose of a matrix X. The first and second index
ranges of the result are X'Range(2) and X'Range(1) respectively.

Each operation returns the result of applying the corresponding operation of
type Real to each component of Left and the matching component of Right.
The index ranges of the result are those of Left. The exception
Constraint_Error is raised if Left'Length(1) is not equal to Right'Length(1) or
Left'Length(2) is not equal to Right'Length(2).

This operation provides the standard mathematical operation for matrix
multiplication. The first and second index ranges of the result are
Left'Range(1) and Right'Range(2) respectively. The exception
Constraint_Error is raised if Left'Length(2) is not equal to Right'Length(1).
This operation involves inner products.

This operation returns the outer product of a (column) vector Left by a
(row) vector Right using the appropriate operation "*" of the type Real for
computing the individual components. The first and second index ranges of the
matrix result are Left'Range and Right'Range respectively.

This operation provides the standard mathematical operation for multiplication
of a (row) vector Left by a matrix Right. The index range of the (row) vector
result is Right'Range(2). The exception Constraint_Error is raised if
Left'Length is not equal to Right'Length(1). This operation involves inner
products.

This operation provides the standard mathematical operation for multiplication
of a matrix Left by a (column) vector Right. The index range of the (column)
vector result is Left'Range(1). The exception Constraint_Error is raised if
Left'Length(2) is not equal to Right'Length. This operation involves inner
products.

This operation returns the result of multiplying each component of Right by the
scalar Left using the "*" operation of the type Real. The index ranges of the
matrix result are those of Right.

Each operation returns the result of applying the corresponding operation of
the type Real to each component of Left and to the scalar Right. The index
ranges of the matrix result are those of Left.

This function returns a vector Y such that X is (nearly) equal to A * Y. This
is the standard mathematical operation for solving a single set of linear
equations. The index range of the result is X'Range. Constraint_Error is
raised if A'Length(1), A'Length(2) and X'Length are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.

This function returns a matrix Y such that X is (nearly) equal to A * Y. This
is the standard mathematical operation for solving several sets of linear
equations. The index ranges of the result are those of X. Constraint_Error
is raised if A'Length(1), A'Length(2) and X'Length(1) are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.

This function returns a matrix B such that A * B is (nearly) the unit matrix.
The index ranges of the result are those of A. Constraint_Error is raised if
A'Length(1) is not equal to A'Length(2). Constraint_Error is raised if
the matrix A is ill-conditioned.

This function returns the determinant of the matrix A. Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2).

This function returns the eigenvalues of the symmetric matrix A as a vector
sorted into order with the largest first. The exception Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2). The index range of the
result is A'Range(1). The exception Argument_Error is raised if the matrix A
is not symmetric.

This procedure computes both the eigenvalues and eigenvectors of the symmetric
matrix A. The out parameter Values is the same as that obtained by calling the
function Eigenvalues. The out parameter Vectors is a matrix whose columns are
the eigenvectors of the matrix A. The order of the columns corresponds to the
order of the eigenvalues. The eigenvectors are normalized and mutually
orthogonal (they are orthonormal), including when there are repeated
eigenvalues. The exception Constraint_Error is raised if A'Length(1) is not
equal to A'Length(2). The index ranges of the parameter Vectors are those of A.
The exception Argument_Error is raised if the matrix A is not symmetric.

This function returns a square "unit matrix" with Order**2 components and
lower bounds of First_1 and First_2 (for the first and second index ranges
respectively). All components are set to 0.0 except for the main diagonal,
whose components are set to 1.0. The exception Constraint_Error is raised if
First_1 + Order - 1 > Integer'Last or First_2 + Order - 1 > Integer'Last.

Accuracy requirements for the subprograms Solve, Inverse, Determinant,
Eigenvalues and Eigensystem are implementation defined.

For operations not involving an inner product, the accuracy
requirements are those of the corresponding operations of the type Real in
both the strict mode and the relaxed mode (see G.2).

For operations involving an inner product, no requirements are specified in
the relaxed mode. In the strict mode the modulus of the absolute error of
the inner product X*Y shall not exceed g*abs(X)*abs(Y) where g is defined as

g = X'Length * Machine_Radix**(1-Machine_Mantissa)

Implementations shall document any techniques used to reduce cancellation
errors such as extended precision arithmetic.

The nongeneric equivalent packages may, but need not, be actual
instantiations of the generic package for the appropriate predefined type.

Implementations should implement the Solve and Inverse functions using
established techniques such as LU decomposition with row interchanges followed
by back and forward substitution. Implementations are recommended to refine the
result by performing an iteration on the residuals; if this is done then it
should be documented.

It is not the intention that any special provision should be made to
determine whether a matrix is ill-conditioned or not. The naturally occurring
overflow (including division by zero) which will result from executing these
functions with an ill-conditioned matrix and thus raise Constraint_Error is
sufficient.

The test that a matrix is symmetric may be performed by using the equality
operator to compare the relevant components.

!corrigendum G.3.2(01)

Insert new clause:

The generic library package Numerics.Generic_Complex_Arrays has the following
declaration:

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

The library package Numerics.Complex_Arrays is declared pure and defines
the same types and subprograms as Numerics.Generic_Complex_Arrays, except
that the predefined type Float is systematically substituted for Real'Base,
and the Real_Vector and Real_Matrix types exported by Numerics.Real_Arrays
are systematically substituted for Real_Vector and Real_Matrix, and the
Complex type exported by Numerics.Complex_Types is systematically
substituted for Complex, throughout. Nongeneric equivalents for each of
the other predefined floating point types are defined similarly, with the
names Numerics.Short_Complex_Arrays, Numerics.Long_Complex_Arrays, etc.

Two types are defined and exported by Ada.Numerics.Generic_Complex_Arrays.
The composite type Complex_Vector is provided to represent a vector with
components of type Complex; it is defined as an unconstrained
one-dimensional array with an index of type Integer. The composite type
Complex_Matrix is provided to represent a matrix with components of type
Complex; it is defined as an unconstrained, two-dimensional array with
indices of type Integer.

The effect of the various subprograms is as described below. In many cases
they are described in terms of corresponding scalar operations in
Numerics.Generic_Complex_Types. Any exception raised by those operations is
propagated by the array subprogram. Moreover any constraints on the parameters
and the accuracy of the result for each individual component is as defined for
the scalar operation.

In the case of those operations which are defined to involve an inner product,
Constraint_Error may be raised if an intermediate result has a component
outside the range of Real'Base even though the final mathematical result
would not.

Each function returns a vector of the specified cartesian components of X.
The index range of the result is X'Range.

Each procedure replaces the specified (cartesian) component of each of
the components of X by the value of the matching component of Re or Im;
the other (cartesian) component of each of the components is unchanged.
The exception Constraint_Error is raised if X'Length is not equal to
Re'Length or Im'Length.

Each function constructs a vector of Complex results (in cartesian
representation) formed from given vectors of cartesian components; when
only the real components are given, imaginary components of zero are assumed.
The index range of the result is Re'Range. The exception Constraint_Error is
raised if Re'Length is not equal to Im'Length.

Each function calculates and returns a vector of the specified polar
components of X or Right using the corresponding function in
Numerics.Generic_Complex_Types. The index range of the result is X'Range or
Right'Range.

Each function constructs a vector of Complex results (in cartesian
representation) formed from given vectors of polar components using the
corresponding function in Numerics.Generic_Complex_Types on matching
components of Modulus and Argument. The index range of the result is
Modulus'Range. The exception Constraint_Error is raised if
Modulus'Length is not equal to Argument'Length.

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Right. The index
range of the result is Right'Range.

This function returns the result of applying the appropriate function
Conjugate in Numerics.Generic_Complex_Types to each component of X. The
index range of the result is X'Range.

Each operation returns the result of applying the corresponding operation
in Numerics.Generic_Complex_Types to each component of Left and the
matching component of Right. The index range of the result is Left'Range.
The exception Constraint_Error is raised if Left'Length is not equal to
Right'Length.

Each operation returns the inner product of Left and Right. The exception
Constraint_Error is raised if Left'Length is not equal to Right'Length.
These operations involve an inner product.

This operation returns the result of multiplying each component of Right by
the complex number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index range of the result is Right'Range.

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the vector Left and the
complex number Right. The index range of the result is Left'Range.

This operation returns the result of multiplying each component of Right by
the real number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index range of the result is Right'Range.

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the vector Left and the
real number Right. The index range of the result is Left'Range.

This function returns a "unit vector" with Order components and a lower
bound of First. All components are set to (0.0,0.0) except for the Index
component which is set to (1.0,0.0). The exception Constraint_Error is
raised if Index < First, Index > First + Order - 1, or
if First + Order - 1 > Integer'Last.

Each function returns a matrix of the specified cartesian components of X.
The index ranges of the result are those of X.

Each procedure replaces the specified (cartesian) component of each of the
components of X by the value of the matching component of Re or Im; the other
(cartesian) component of each of the components is unchanged. The exception
Constraint_Error is raised if X'Length(1) is not equal to Re'Length(1) or
Im'Length(1) or if X'Length(2) is not equal to Re'Length(2) or Im'Length(2).

Each function constructs a matrix of Complex results (in cartesian
representation) formed from given matrices of cartesian components; when
only the real components are given, imaginary components of zero are
assumed. The index ranges of the result are those of Re. The exception
Constraint_Error is raised if Re'Length(1) is not equal to Im'Length(1) or
Re'Length(2) is not equal to Im'Length(2).

Each function calculates and returns a matrix of the specified polar
components of X or Right using the corresponding function in
Numerics.Generic_Complex_Types. The index ranges of the result are those of X
or Right.

Each function constructs a matrix of Complex results (in cartesian
representation) formed from given matrices of polar components using
the corresponding function in Numerics.Generic_Complex_Types on matching
components of Modulus and Argument. The index ranges of the result are those
of Modulus. The exception Constraint_Error is raised if Modulus'Length(1) is
not equal to Argument'Length(1) or Modulus'Length(2) is not equal to
Argument'Length(2).

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Right. The index
ranges of the result are those of Right.

This function returns the result of applying the appropriate function Conjugate
in Numerics.Generic_Complex_Types to each component of X. The index ranges of
the result are those of X.

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of Left and the matching
component of Right. The index ranges of the result are those of
Left. The exception Constraint_Error is raised if Left'Length(1) is not equal to
Right'Length(1) or Left'Length(2) is not equal to Right'Length(2).

This operation returns the outer product of a (column) vector Left by a (row)
vector Right using the appropriate operation "*" in
Numerics.Generic_Complex_Types for computing the individual components.
The first and second index ranges of the matrix result are Left'Range and
Right'Range respectively.

Each operation provides the standard mathematical operation for matrix
multiplication. The first and second index ranges of the result are
Left'Range(1) and Right'Range(2) respectively. The exception
Constraint_Error is raised if Left'Length(2) is not equal to Right'Length(1).
These operations involve inner products.

Each operation returns the outer product of a (column) vector Left by a (row)
vector Right using the appropriate operation "*" in
Numerics.Generic_Complex_Types for computing the individual components.
The first and second index ranges of the matrix result are Left'Range and
Right'Range respectively.

Each operation provides the standard mathematical operation for multiplication
of a (row) vector Left by a matrix Right. The index range of the (row) vector
result is Right'Range(2). The exception Constraint_Error is raised if
Left'Length is not equal to Right'Length(1). These operations involve inner
products.

Each operation provides the standard mathematical operation for multiplication
of a matrix Left by a (column) vector Right. The index range of the (column)
vector result is Left'Range(1). The exception Constraint_Error is raised if
Left'Length(2) is not equal to Right'Length. These operations involve inner
products.

This operation returns the result of multiplying each component of Right by
the complex number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index ranges of the result are those of
Right.

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the matrix Left and the
complex number Right. The index ranges of the result are those of Left.

This operation returns the result of multiplying each component of Right by
the real number Left using the appropriate operation "*" in
Numerics.Generic_Complex_Types. The index ranges of the result are those of
Right.

Each operation returns the result of applying the corresponding operation in
Numerics.Generic_Complex_Types to each component of the matrix Left and the
scalar Right. The index ranges of the result are those of Left.

This function returns the eigenvalues of the Hermitian matrix A as a vector
sorted into order with the largest first. The exception Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2). The index range of the
result is A'Range(1). The exception Argument_Error is raised if the matrix A
is not Hermitian.

This procedure computes both the eigenvalues and eigenvectors of the Hermitian
matrix A. The out parameter Values is the same as that obtained by calling the
function Eigenvalues. The out parameter Vectors is a matrix whose columns are
the eigenvectors of the matrix A. The order of the columns corresponds to the
order of the eigenvalues. The eigenvectors are mutually orthonormal,
including when there are repeated eigenvalues. The exception Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2). The index ranges of the
parameter Vectors are those of A. The exception Argument_Error is raised if the
matrix A is not Hermitian.

This function returns a square "unit matrix" with Order**2 components and
lower bounds of First_1 and First_2 (for the first and second index ranges
respectively). All components are set to (0.0,0.0) except for the main diagonal,
whose components are set to (1.0,0.0). The exception Constraint_Error is raised
if First_1 + Order - 1 > Integer'Last or First_2 + Order - 1 > Integer'Last.

For operations not involving an inner product, the accuracy requirements are
those of the corresponding operations of the type Real'Base and Complex in both
the strict mode and the relaxed mode (see G.2).

g = sqrt(2.0) * X'Length * Machine_Radix**(1-Machine_Mantissa) for mixed
complex and real operands

g = sqrt(8.0) * X'Length * Machine_Radix**(1-Machine_Mantissa) for two
complex operands

Although many operations are defined in terms of operations from
Numerics.Generic_Complex_Types, they need not be implemented by calling those
operations provided that the effect is the same.

Implementations should implement the Solve and Inverse functions using
established techniques. Implementations are recommended to refine the result by
performing an iteration on the residuals; if this is done then it should be
documented.

The test that a matrix is Hermitian may use the equality operator to compare
the real components and negation followed by equality to compare the imaginary
components (see G.2.1).

Implementations should not perform operations on mixed complex and real operands
by first converting the real operand to complex. See G.1.1.

!ACATS Test

ACATS test(s) need to be created.

!appendix

**************************************************************** Recommendation on ISO/IEC 13813 from the UK The standard ISO/IEC 13813 entitled generic packages of real and complex type declarations and basic operations for Ada (including vector and matrix types) will soon be up for review. This note reviews the background to the development of the standard and makes a recommendation that the standard be revised. Background The Numerics Working Group of WG9 met many times during the period when Ada 95 was being designed and produced a number of standards. They were faced with the problem of whether to produce standards based on Ada 83 (87 in ISO terms) or whether to base them on Ada 95 or subsume them into Ada 95. One dilemma was of course that although Ada 95 was on the way nevertheless Ada 83 was expected to continue in use for many years. The standards are 11430: Generic package of elementary functions for Ada. 11729: Generic package of primitive functions for Ada. 13813: Generic packages of real and complex type declarations and basic operations for Ada (including vector and matrix types). 13814: Generic package of complex elementary functions for Ada. 11430 and 11729 are mentioned for completeness. They were published in 1994. They were based entirely on Ada 83 and their facilities are provided in the Ada 95 core language. The elementary functions, 11430, became the package Ada.Numerics.Generic_Elementary_Functions and the primitive functions, 11729, became the various attributes such as 'Floor and 'Ceiling, and 'Exponent and 'Fraction. These two standards were withdrawn recently and need no further mention. The other two standards, 13813 and 13814, were published in 1998 and will soon be up for review at the end of their five year period. Three possible fates can befall a standard when it is reviewed. It can be withdrawn, revised or confirmed. In the case of 13814, the functionality is all incorporated into the Numerics Annex of Ada 95 as the package Ada.Numerics.Generic_Complex_Elementary_Functions. There are a few changes in presentation because the Ada 95 package uses the generic package parameter feature which of course did not exist in Ada 83. Nevertheless there seems little point in continuing with 13814 and so at the Leuven meeting of WG9 it was agreed to recommend that it be withdrawn. However, the situation regarding 13813 is not so clear. Some of its functionality is included in Ada 95 but quite a lot is not. The topics covered are (1) a complex types package including various complex arithmetic operations, (2) a real arrays package covering both vectors and matrices, (3) a complex arrays package covering both vectors and matrices, (4) a complex input-output package. The complex types package (1) became the package Ada.Numerics.Generic_Complex_Types and the input-output package (4) became Ada.Text_IO.Complex_IO. However, the array packages, both real and complex, were not incorporated into the Ada 95 standard. At the Leuven meeting, it was agreed that 13813 should not be withdrawn without further study. The UK was asked to study whether small or large changes are required in 13813 and to report back. The Ada Rapporteur Group would then decide whether the functionality should be included in a future revision or amendment to Ada 95. This is the report from the UK. Recommendation It is recommended that 13813 be revised so that it only contains the functionality not included in Ada 95. The revised standard should contain two generic packages namely Ada.Numerics.Generic_Real_Arrays and Ada.Numerics.Generic_Complex_Arrays. There should also be standard non-generic packages corresponding to the predefined types such as Float in an analogous manner to the standard packages such as Ada.Numerics.Complex_Types and Ada.Numerics.Long_Complex_Types for Float and Long_Float respectively. The text of the Ada specifications of the two generic packages should be essentially as given in the nonnormative Annex G of the existing standard 13813. (This Annex illustrates how the existing standard packages might be rewritten using Ada 95. There is an error regarding the formal package parameters which has been corrected in the revised text.) There is an important issue regarding what should happen if there is a mismatch in the array lengths of the parameters in a number of the subprograms provided by the packages. For example if function "+" (Left, Right: Real_Vector) return Real_Vector be called with parameters such that Left'Length /= Right'Length. The existing standard raises the exception Array_Index_Error which is declared alone in a package Array_Exceptions. The nonnormative Annex G shows this exception incorporated into the package Ada.Numerics thereby producing an incompatibility with the existing definition of Ada.Numerics. We considered four possibilities regarding this exception 1) Add Array_Index_Error to Ada.Numerics as in Annex G. 2) Place Array_Index_Error in a new child package such as Ada.Numerics.Arrays. 3) Eliminate Array_Index_Error and raise Constraint_Error instead. 4) State that the behaviour with mismatched arrays is implementation-defined. We concluded that Option (1) is undesirable because of incompatibility. Option (2) is feasible but one ought then to place the generic packages themselves into this package so that they become Ada.Numerics.Arrays.Generic_Real_Arrays and Ada.Numerics.Arrays.Generic_Complex_Arrays. This nesting is considered cumbersome. Option(3) gives the same behaviour as similar mismatching on predefined operations and although losing some specificacity has practical simplicity. Option (4) is disliked since gratuitous implementation-defined behaviour should be avoided. We therefore recommend Option (3) that Constraint_Error be raised on mismatching of parameters. If the Ada95 standard itself be revised at some later date then consideration should be given to incorporating the functionality of the revised 13813 into the Numerics Annex. Proposed text The proposed normative text of the revised standard is distributed separately as N404, Working Draft, Revision of ISO/IEC 13813. Note that the non-normative rationale section remains to be completed and that consequently the bibliography might need alterations to match. Acknowledgment We acknowledge the valuable assistance of Donald Sando, the editor of the original standard, in the preparation of this recommendation. **************************************************************** From: John Barnes Sent: Tuesday, January 21, 2003 10:33 AM Gosh - I have made a start on AI-296. [Editor's note: This is version /01 of the AI.] I have pulled all the Ada text in and written a first cut at the description for the real vectors and matrices. I thought maybe it would be a good idea to get the style settled before spending time on the complex ones. Here are some thoughts. There are questions of how to arrange the annex in !proposal. What do we use for the plural of index when talking about arrays? I suspect that it ought to be indexes and not indices as Don had. I have slimmed down the accuracy and error stuff that Don Sando had by simply referring to the underlying real operations. There are as yet no useful remarks on the accuracy of inner product. I started by explaining the bounds of the result in terms of the bounds of the parameters whereas Don had done it in terms of ranges. Later I discovered that ranges are much easier for the more elaborate matrix cases so perhaps I should have used ranges throughout. Thus saying for example "the index range of the result is Left'Range" rather than "the bounds of the result are those of the left operand". I am not sure whether I need to say anything about null arrays. I note that 4.5.1(8) doesn't seem to. Incidentally, I am still overwhelmed with other work. Although the Spark book has gone away to be proof read, it re-emerges tomorrow and will keep me busy for several days in doing final corrections. Also I have to prepare a one-day course on Spark at the University of York the week after Padua so I have to send the notes beforehand. I do this course each year but at the last minute they want a lot of changes. But we do have a baseline to discuss on AI-296 for Padua even if I don't get a chance to spend much more time on it before then. So I don't feel too guilty. There is a lot of interest in this stuff (especially problems of accuracy) in the UK and we have a BSI meeting in mid February to discuss this. **************************************************************** From: Pascal Leroy Sent: Wednesday, January 22, 2003 8:24 AM > There are as yet no useful remarks on the > accuracy of inner product. I think that the accuracy of the inner product should be defined in a way similar to that of the complex multiplication. If you look at the inner product of vectors (a1, a2) and (b1, b2), the result is quite similar to the real part of the (complex) multiplication of a1 + ib1 and a1 - ib2. The box error in G.2.5 ensures that if the result of a multiplication lands very close to one of the axis, you don't have to provide an unreasonable accuracy on the "small" component, because presumably cancellations happened. Moreover, the "large" component gives you some information on the magnitude of the cancellation, so it can be used to define the acceptable error. That's essentially what the box error does (at least that's my understanding). For the inner product of v1 and v2, I believe that the error should similarly be defined to be of the form d * length(v1) * length (v2). If cancellations happen, i.e. if the vectors are nearly orthogonal, the error can be quite large compared to the final result. In the absence of cancellations, i.e. if the vectors are not orthogonal, then with a proper choice for d we can require good accuracy. The alert reader will notice that this amounts to defining the relative error on the result as d/cos (A) where A is the angle of the two vectors. This is just a back-of-an-envelope proposal. No rigorous analysis was done. **************************************************************** From: John Barnes Sent: Tuesday, June 10, 2003 4:42 AM Vectors and matrices At the last ARG meeting I was asked to add material for matrix inversion, solution of linear equations, determinants, and eigenvalues and vectors. We very briefly considered a couple of other topics but dismissed them as tricky. I have updated AI-296 as instructed and sent it to Randy so I assume it is now on the database. [Editor's note: This is version /02 of the AI.] I put the linear equations, invert and determinant in the main packages (one for reals and one for complex). I put the eigenvalues and vectors in child packages because they seemed a bit different - also they were more complex than I had remembered (I hadn't looked at this stuff for nearly half a century) and maybe they are more elaborate than we need. There are a number of queries embedded in the text concerning accuracy etc. However, there are a number of other matters which ought to be discussed. I visited NAG (National Algorithms Group). They did a lot of Ada 83 numerics stuff once although they no longer sell it. They did implement 13813. They put the invert stuff in separate packages but it is much more elaborate than my humble proposal. For example they have options on whether to invert just once or to iterate on the residuals until they are acceptable. But then of course one has to give the accuracy required (although they do have defaults). Moreover the LU decomposition is made visible. Now as a user, I don't really want to know about how it is done (in a sense it breaks the abstraction), I just want the answer. They have provided what the numerical analyst specialist would want and I don't really think that is appropriate for a simple standard but maybe I am wrong. Thus their subprograms have some 10 parameters whereas in my proposal they are the minimal one or two. For instance their determinant has eight parameters whereas mine has one. In the case of eigenvalues and vectors, they had options to select just the largest or smallest or all those in a certain range. No doubt because of timing considerations for large matrices. We also considered other topics which might provide candidates for a standard. The trouble with many topics is that techniques are still evolving and it is unclear that a unique worthy proposal would be easy. Curve and surface fitting. There is lots of choice here. Cubic splines are fashionable. Chebychev polynomials are a long established technique. But both require significant user knowledge and it is easy to misuse them. Maximization/minimization. Lots of options here and real problems have constraints. The techniques are heuristic and still evolving so this looks really hard and can be eliminated. Partial differential equations seem too much for the casual user. In any event there is a huge variety of parameters and constraints. Ordinary differential equations and quadrature (integration) are perhaps more promising. Again there are quite a lot of options. But this might be worth exploring. Roots of general non-linear equations. An evolving subject still. Very heuristic and prone to chaotic difficulties. Roots of polynomial equations. This seemed more likely . It appears that the subject is static and a well-established technique devised by Laguerre is typically used. (But it is outside my domain of knowledge.) I have also contemplated my 20 year old HP pocket calculator. I had a vague feeling that Ada should be able to do what it can do. It can do the following: The usual trigonometric and hyperbolic functions. Real and complex matrices, linear equations, inversion and determinants (reveals the LU decomposition). It does not do eigenvalues. Random number generation. A whole heap of statistical stuff. Integration (quadrature) of a function of one variable. Solution of a non-linear equation in one variable. So where do we go from here? It is a slippery slope and I fear that I am not an expert. **************************************************************** From: John Barnes Sent: Thursday, July 24, 2003 2:09 AM Version /03 is as /02 dated 03-05-22 but has vector product removed and other minor changes agreed at the ARG meeting in Toulouse. It does not include LU decomposition. **************************************************************** From: John Barnes Sent: Thursday, July 24, 2003 2:09 AM Version /04 is as /03 dated 03-07-22 but includes LU decomposition. **************************************************************** From: John Barnes Sent: Friday, July 25, 2003 2:29 AM I have created two new versions of this AI and Randy has put them on the database (he tells me I had some spurious CRs which he had to edit and so he warns that perhaps they are not perfect in that respect). The first (version 03) incorporates various minor changes agreed at Toulouse but does not include LU decomposition. The second (version 04) includes LU decomposition. The ZIP file only contains 04. Randy tells me so you will have to go to the database directly for 03. (I always use the database directly so maybe that's no problem.) You may recall (or then you may not) that LU decomposition enables a square matrix to be held in a form which is faster for operations such as invert, solve and eigenvalues. It is of value when several operations have to be performed on the same matrix because time is saved by not repeating some common operations. This was important when hardware was slow. But hardware is so fast now that it can hardly matter in most circumstances. Perhaps it might matter in a tight loop when the same matrix is being used to solve a set of more than 100 equations repeatedly. But does this ever happen? Seems doubtful and if it did a real numerical analyst needs to be called in anyway. Note that large banded matrices do occur in differential equations but that is specialized stuff anyway which we are not addressing. Moreover adding LU adds complexity to the RM and an irritating visibility problem explained in the AI discussion. Since our goal here I believe is to provide some simple useful facilities for the common programmer (while still providing a base which can be used through child packages to provide other stuff) it all now seems too much to me. As a programmer nearly half a century ago, I used to invert lots of smallish matrices and never used LU decomposition (maybe it hadn't been discovered/invented). Maybe there is some partial analogy with all those wonderful algorithms for dealing with large matrices held on several magnetic tapes. All in the past surely. Of course LU has some nice mathematics behind it but not every Ada programmer needs to be exposed to such things. I am sure that any professional numerical analyst will say that LU is a must have. But that could be more out of professional pride, habit and rectitude rather than practical need. I am also sure that if presented with the sort of numerical problems I dabble with from time to time I would never bother with the LU forms because the straightforward ones would do the job adequately. Another point is that the existence of the LU forms might make life harder for software maintenance by obscuring what is going on. Perhaps I should add that my colleagues in the UK who have looked at this are moving to the view that LU is unnecessary. And discussions with Pascal before he went on vacation were going that way as well. We can discuss it in Sydney. But any thoughts before then might be welcome. I suppose I will have to do the wording for downward closures now - groan. **************************************************************** From: John Barnes Sent: Tuesday, September 2, 2003 6:40 AM This version [version /05 - ED] is as 02 dated 03-05-22 but has vector product removed and other minor changes agreed at the ARG meeting in Toulouse. It does not include LU decomposition. Moreover, the eigenvalues and vectors of nonsymmetric, non-Hermitian matrices have been removed because of potential computational difficulties. **************************************************************** From: John Barnes Sent: Tuesday, September 2, 2003 6:54 AM I have had a number of further discussions on this topic with people at NAG, Farnborough, Pascal etc. I have also invested in a heavy tome on numerical stuff. As a consequence I have concluded that making LU decomposition visible to the user was unnecessary. Moreover, after reading around, it seems that some of the eigenvalue calculations can be tricky and accordingly I have reduced the complex package to the one straightforward case. I attach a paper written for the Ada User Journal which explains this in more detail. And so I have written yet another version of this AI with which I feel fairly comfortable. This is version 5 dated 2003/09/01 and I have sent this to Randy for placing on the database. For those who cannot wait I also attach it. For those of a historical inclination, the versions are as follows 02 as presented at Toulouse 03 as 02 plus some minor corrections agreed at Toulouse 04 as 03 plus LU decomposition 05 as 03 minus eigensystems of general matrices **************************************************************** From: Martin Dowie Sent: Monday, September 22, 2003 8:41 AM Small correction to the spec as given in the latest AI: Currently: generic package Ada.Numerics.Generic_Complex_Arrays.Eigen is ... Should be: with Ada.Numerics.Generic_Real_Arrays; generic with package Real_Arrays is new Ada.Numerics.Generic_Real_Arrays (<>); use Real_Arrays; package Ada.Numerics.Generic_Complex_Arrays.Eigen is ... **************************************************************** From: Pascal Leroy Sent: Monday, September 22, 2003 8:41 AM Eigen is a child of Generic_Complex_Arrays, and Generic_Complex_Arrays has a generic formal parameter Real_Arrays, so the child has access to that parameter. There is no need to repeat it. **************************************************************** From: Martin Dowie Sent: Monday, September 22, 2003 11:02 AM Must be a GNAT (3.15p) bug then... :-( ObjectAda (Win32) doesn't seem to support Annex G, so I couldn't cross-check... ****************************************************************

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