!standard G (00) 03-09-01 AI95-00296/05
!class amendment 02-06-07
!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
operations for the solution of linear equations, matrix inversion and
determinants. Furthermore, child packages provide subprograms for 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.
Generic child packages for the determination of the eigenvalues and
eigenvectors of real symmetric matrices and Hermitian matrices are defined
in G.3.3.
G.3.1 Real Vectors and Matrices
The generic library package Numerics.Generic_Real_Arrays has the following
declaration:
generic
type Real is digits <>;
package Ada.Numerics.Generic_Real_Arrays is
pragma Pure(Generic_Real_Arrays);
-- Types
type Real_Vector is array (Integer range <>) of Real'Base;
type Real_Matrix is array (Integer range <>, Integer range <>) of Real'Base;
-- Subprograms for Real_Vector types
-- Real_Vector arithmetic operations
function "+" (Right : Real_Vector) return Real_Vector;
function "-" (Right : Real_Vector) return Real_Vector;
function "abs" (Right : Real_Vector) return Real_Vector;
function "+" (Left, Right : Real_Vector) return Real_Vector;
function "-" (Left, Right : Real_Vector) return Real_Vector;
function "*" (Left, Right : Real_Vector) return Real'Base;
-- Real_Vector scaling operations
function "*" (Left : Real'Base; Right : Real_Vector) return Real_Vector;
function "*" (Left : Real_Vector; Right : Real'Base) return Real_Vector;
function "/" (Left : Real_Vector; Right : Real'Base) return Real_Vector;
-- Other Real_Vector operations
function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Real_Vector;
-- Subprograms for Real_Matrix types
-- Real_Matrix arithmetic operations
function "+" (Right : Real_Matrix) return Real_Matrix;
function "-" (Right : Real_Matrix) return Real_Matrix;
function "abs" (Right : Real_Matrix) return Real_Matrix;
function Transpose (X : Real_Matrix) return Real_Matrix;
function "+" (Left, Right : Real_Matrix) return Real_Matrix;
function "-" (Left, Right : Real_Matrix) return Real_Matrix;
function "*" (Left, Right : Real_Matrix) return Real_Matrix;
function "*" (Left, Right : Real_Vector) return Real_Matrix;
function "*" (Left : Real_Vector; Right : Real_Matrix) return Real_Vector;
function "*" (Left : Real_Matrix; Right : Real_Vector) return Real_Vector;
-- Real_Matrix scaling operations
function "*" (Left : Real'Base; Right : Real_Matrix) return Real_Matrix;
function "*" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix;
function "/" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix;
-- Real_Matrix inversion and related operations
function Solve (A : Real_Matrix; X: Real_Vector) return Real_Vector;
function Solve (A, X : Real_Matrix) return Real_Matrix;
function Inverse (A : Real_Matrix) return Real_Matrix;
function Determinant (A : Real_Matrix) return Real'Base;
-- Other Real_Matrix operations
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Real_Matrix;
private
-- implementation-defined
end Ada.Numerics.Generic_Real_Arrays;
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.
function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Real_Vector;
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.
function Transpose (X : Real_Matrix) return Real_Matrix;
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.
function Solve (A : Real_Matrix; X: Real_Vector) return Real_Vector;
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.
function Solve (A, X : Real_Matrix) return Real_Matrix;
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.
function Inverse (A : Real_Matrix) return Real_Matrix;
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.
function Determinant (A : Real_Matrix) return Real'Base;
This function returns the determinant of the matrix A. Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2).
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1) return Real_Matrix;
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 functions Solve, Inverse and Determinant 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 accuracy should be at least that of
the canonical implementation of multiplication and addition using the
corresponding operations of type Real'Base and performing the cumulative
addition using ascending indices. Implementations shall document any
techniques used to reduce cancellation errors such as extended precision
arithmetic.
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 may choose 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.
G.3.2 Complex Vectors and Matrices
The generic library package Numerics.Generic_Complex_Arrays has the following
declaration:
with Ada.Numerics.Generic_Real_Arrays, Ada.Numerics.Generic_Complex_Types;
generic
with package Real_Arrays is new Ada.Numerics.Generic_Real_Arrays (<>);
use Real_Arrays;
with package Complex_Types is new Ada.Numerics.Generic_Complex_Types (Real);
use Complex_Types;
package Ada.Numerics.Generic_Complex_Arrays is
pragma Pure(Generic_Complex_Arrays);
-- Types
type Complex_Vector is array (Integer range <>) of Complex;
type Complex_Matrix is array (Integer range <>,
Integer range <>) of Complex;
-- Subprograms for Complex_Vector types
-- Complex_Vector selection, conversion and composition operations
function Re (X : Complex_Vector) return Real_Vector;
function Im (X : Complex_Vector) return Real_Vector;
procedure Set_Re (X : in out Complex_Vector;
Re : in Real_Vector);
procedure Set_Im (X : in out Complex_Vector;
Im : in Real_Vector);
function Compose_From_Cartesian (Re : Real_Vector) return Complex_Vector;
function Compose_From_Cartesian (Re, Im : Real_Vector) return Complex_Vector;
function Modulus (X : Complex_Vector) return Real_Vector;
function "abs" (Right : Complex_Vector) return Real_Vector
renames Modulus;
function Argument (X : Complex_Vector) return Real_Vector;
function Argument (X : Complex_Vector;
Cycle : Real'Base) return Real_Vector;
function Compose_From_Polar (Modulus, Argument : Real_Vector)
return Complex_Vector;
function Compose_From_Polar (Modulus, Argument : Real_Vector;
Cycle : Real'Base)
return Complex_Vector;
-- Complex_Vector arithmetic operations
function "+" (Right : Complex_Vector) return Complex_Vector;
function "-" (Right : Complex_Vector) return Complex_Vector;
function Conjugate (X : Complex_Vector) return Complex_Vector;
function "+" (Left, Right : Complex_Vector) return Complex_Vector;
function "-" (Left, Right : Complex_Vector) return Complex_Vector;
function "*" (Left, Right : Complex_Vector) return Complex;
-- Mixed Real_Vector and Complex_Vector arithmetic operations
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;
function "*" (Left : Real_Vector; Right : Complex_Vector) return Complex;
function "*" (Left : Complex_Vector; Right : Real_Vector) return Complex;
-- Complex_Vector scaling operations
function "*" (Left : Complex;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Complex) return Complex_Vector;
function "/" (Left : Complex_Vector;
Right : Complex) return Complex_Vector;
function "*" (Left : Real'Base;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Real'Base) return Complex_Vector;
function "/" (Left : Complex_Vector;
Right : Real'Base) return Complex_Vector;
-- Other Complex_Vector operations
function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Complex_Vector;
-- Subprograms for Complex_Matrix types
-- Complex_Matrix selection, conversion and composition operations
function Re (X : Complex_Matrix) return Real_Matrix;
function Im (X : Complex_Matrix) return Real_Matrix;
procedure Set_Re (X : in out Complex_Matrix;
Re : in Real_Matrix);
procedure Set_Im (X : in out Complex_Matrix;
Im : in Real_Matrix);
function Compose_From_Cartesian (Re : Real_Matrix) return Complex_Matrix;
function Compose_From_Cartesian (Re, Im : Real_Matrix) return Complex_Matrix;
function Modulus (X : Complex_Matrix) return Real_Matrix;
function "abs" (Right : Complex_Matrix) return Real_Matrix
renames Modulus;
function Argument (X : Complex_Matrix) return Real_Matrix;
function Argument (X : Complex_Matrix;
Cycle : Real'Base) return Real_Matrix;
function Compose_From_Polar (Modulus, Argument : Real_Matrix)
return Complex_Matrix;
function Compose_From_Polar (Modulus, Argument : Real_Matrix;
Cycle : Real'Base)
return Complex_Matrix;
-- Complex_Matrix arithmetic operations
function "+" (Right : Complex_Matrix) return Complex_Matrix;
function "-" (Right : Complex_Matrix) return Complex_Matrix;
function Conjugate (X : Complex_Matrix) return Complex_Matrix;
function Transpose (X : Complex_Matrix) return Complex_Matrix;
function "+" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "-" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left, Right : Complex_Vector) return Complex_Matrix;
function "*" (Left : Complex_Vector;
Right : Complex_Matrix) return Complex_Vector;
function "*" (Left : Complex_Matrix;
Right : Complex_Vector) return Complex_Vector;
-- Mixed Real_Matrix and Complex_Matrix arithmetic operations
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;
function "*" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
function "*" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Matrix;
function "*" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Matrix;
function "*" (Left : Real_Vector;
Right : Complex_Matrix) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Real_Matrix) return Complex_Vector;
function "*" (Left : Real_Matrix;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Matrix;
Right : Real_Vector) return Complex_Vector;
-- Complex_Matrix scaling operations
function "*" (Left : Complex;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Complex) return Complex_Matrix;
function "/" (Left : Complex_Matrix;
Right : Complex) return Complex_Matrix;
function "*" (Left : Real'Base;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Real'Base) return Complex_Matrix;
function "/" (Left : Complex_Matrix;
Right : Real'Base) return Complex_Matrix;
-- Complex_Matrix inversion and related operations
function Solve (A : Complex_Matrix; X: Complex_Vector) return Complex_Vector;
function Solve (A, X : Complex_Matrix) return Complex_Matrix;
function Inverse (A : Complex_Matrix) return Complex_Matrix;
function Determinant (A : Complex_Matrix) return Complex;
-- Other Complex_Matrix operations
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Complex_Matrix;
end Ada.Numerics.Generic_Complex_Arrays;
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.
function Re (X : Complex_Vector) return Real_Vector;
function Im (X : Complex_Vector) return Real_Vector;
Each function returns a vector of the specified cartesian components of X.
The index range of the result is X'Range.
procedure Set_Re (X : in out Complex_Vector; Re : in Real_Vector);
procedure Set_Im (X : in out Complex_Vector; Im : in Real_Vector);
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.
function Compose_From_Cartesian (Re : Real_Vector) return Complex_Vector;
function Compose_From_Cartesian (Re, Im : Real_Vector) return Complex_Vector;
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.
function Modulus (X : Complex_Vector) return Real_Vector;
function "abs" (Right : Complex_Vector) return Real_Vector renames Modulus;
function Argument (X : Complex_Vector) return Real_Vector;
function Argument (X : Complex_Vector;
Cycle : Real'Base) return Real_Vector;
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.
function Compose_From_Polar (Modulus, Argument : Real_Vector)
return Complex_Vector;
function Compose_From_Polar (Modulus, Argument : Real_Vector; Cycle : Real'Base)
return Complex_Vector;
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.
function Conjugate (X : Complex_Vector) return Complex_Vector;
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.
function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Complex_Vector;
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.
function Re (X : Complex_Matrix) return Real_Matrix;
function Im (X : Complex_Matrix) return Real_Matrix;
Each function returns a matrix of the specified cartesian components of X.
The index ranges of the result are those of X.
procedure Set_Re (X : in out Complex_Matrix; Re : in Real_Matrix);
procedure Set_Im (X : in out Complex_Matrix; Im : in Real_Matrix);
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).
function Compose_From_Cartesian (Re : Real_Matrix) return Complex_Matrix;
function Compose_From_Cartesian (Re, Im : Real_Matrix) return Complex_Matrix;
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).
function Modulus (X : Complex_Matrix) return Real_Matrix;
function "abs" (Right : Complex_Matrix) return Real_Matrix renames Modulus;
function Argument (X : Complex_Matrix) return Real_Matrix;
function Argument (X : Complex_Matrix;
Cycle : Real'Base) return Real_Matrix;
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.
function Compose_From_Polar (Modulus, Argument : Real_Matrix)
return Complex_Matrix;
function Compose_From_Polar (Modulus, Argument : Real_Matrix;
Cycle : Real'Base)
return Complex_Matrix;
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.
function Conjugate (X : Complex_Matrix) return Complex_Matrix;
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.
function Transpose (X : Complex_Matrix) return Complex_Matrix;
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.
function Solve (A : Complex_Matrix; X: Complex_Vector) return Complex_Vector;
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.
function Solve (A, X : Complex_Matrix) return Complex_Matrix;
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.
function Inverse (A : Complex_Matrix) return Complex_Matrix;
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.
function Determinant (A : Complex_Matrix) return Complex;
This function returns the determinant of the matrix A. Constraint_Error is
raised if A'Length(1) is not equal to A'Length(2).
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Complex_Matrix;
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 functions Solve, Inverse and Determinant 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 accuracy should be at least that of
the canonical implementation of multiplication and addition using the
corresponding operations of type Real'Base and Complex and performing the
cumulative addition using ascending indices. Implementations shall document
any techniques used to reduce cancellation errors such as extended precision
arithmetic.
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 may choose 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.
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).
G.3.3 Eigensystems of Matrices
Eigenvalues and eigenvectors of a real symmetric matrix are computed using
subprograms in the child package Ada.Numerics.Generic_Real_Arrays.Eigen
whose declaration is:
generic
package Ada.Numerics.Generic_Real_Arrays.Eigen is
pragma Pure(Eigen);
-- Eigensystem of a real symmetric matrix
function Values(A : Real_Matrix) return Real_Vector;
procedure Values_And_Vectors(A : in Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix);
end Ada.Numerics.Generic_Real_Arrays.Eigen;
The library package Numerics.Real_Arrays.Eigen is declared pure and defines
the same subprograms as Numerics.Generic_Real_Arrays.Eigen, 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.Eigen, Numerics.Long_Real_Arrays.Eigen, etc.
The effect of the subprograms is as described below.
function Values(A : Real_Matrix) return Real_Vector;
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).
procedure Values_And_Vectors(A : in Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix);
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 Values. 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.
For both subprograms, if the matrix A is not symmetric then Argument_Error is
raised.
Eigenvalues and eigenvectors of a Hermitian matrix are computed using
subprograms in the child package Ada.Numerics.Generic_Complex_Arrays.Eigen
whose declaration is:
generic
package Ada.Numerics.Generic_Complex_Arrays.Eigen is
pragma Pure(Eigen);
-- Eigensystem of a Hermitian matrix
function Values(A : Complex_Matrix) return Real_Vector;
procedure Values_And_Vectors(A : in Complex_Matrix;
Values : out Real_Vector;
Vectors : out Complex_Matrix);
end Ada.Numerics.Generic_Complex_Arrays.Eigen;
The library package Numerics.Complex_Arrays.Eigen is declared pure and defines
the same subprograms as Numerics.Generic_Complex_Arrays.Eigen, except that
the Real_Vector and Real_Matrix types exported by Numerics.Real_Arrays are
systematically substituted for Real_Vector and Real_Matrix throughout.
Nongeneric equivalents for each of the other predefined floating point types
are defined similarly, with the names
Numerics.Short_Complex_Arrays.Eigen, Numerics.Long_Complex_Arrays.Eigen, etc.
The effect of the subprograms is as described below.
function Values(A : Complex_Matrix) return Real_Vector;
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).
procedure Values_And_Vectors(A : in Complex_Matrix;
Values : out Real_Vector;
Vectors : out Complex_Matrix);
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 Values. 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.
For both subprograms, if the matrix A is not Hermitian then Argument_Error is
raised.
Implementation Requirements
The accuracy of these subprograms is implementation defined.
Implementation Permissions
The nongeneric equivalent packages may, but need not, be actual
instantiations of the generic package for the appropriate predefined type.
Implementation Advice
The test that a matrix is symmetric may be performed by using the equality
operator to compare the relevant components. The test that a matrix is
Hermitian may similarly use the equality operator to compare the real
components and negation followed by equality to compare the imaginary
components (see G.2.1).
?? I hope that is good enough. I am not sure whether the RM implies that
negation is exact. If it doesn't then the paragraph above needs changing or
G.2.1 needs changing.
!example
!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 to provide something useful for the user who is not a
numerical professional and also 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 simply
required that in the exact mode, the accuracy should be no less than that of
the canonical implementation using a loop statement. This is because on some
hardare, 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.
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
X := X + DX;
Implementations may choose 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 helpful in
deciding whether an array is ill-conditioned and therefore the results of
inversion might be suspect.
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. Because these
are somewhat more specialized than the elementary operations they have been
placed in generic child packages such as Numerics.Generic_Real_Arrays.Eigen.
The individual subprograms are Values and Values_And_Vectors which means that
they can be invoked by statements such as
V := Eigen.Values(A);
There is no separate function 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. The subprograms for these are in the real child
package. 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.
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 and vectors. The
subprograms for these are in the complex child package. 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, determinant evaluation and eigensystems 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.
!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 polynolmials 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
****************************************************************