G.3.1 Real Vectors and Matrices
Static Semantics
The generic library
package Numerics.Generic_Real_Arrays has the following declaration:
generic
type Real
is digits <>;
package Ada.Numerics.Generic_Real_Arrays
with Pure, Nonblocking
is
-- 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;
function "abs" (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;
-- Eigenvalues and vectors of a real symmetric matrix
function Eigenvalues (A : Real_Matrix)
return Real_Vector;
procedure Eigensystem (A :
in Real_Matrix;
Values :
out Real_Vector;
Vectors :
out Real_Matrix);
-- Other Real_Matrix operations
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Real_Matrix;
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 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 subprograms is as described
below. In most cases the subprograms 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. 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. Constraint_Error is raised if Left'Length is not equal to
Right'Length. This operation involves an inner product.
function "abs" (Right : Real_Vector) return Real'Base;
This operation returns the L2-norm of Right (the
square root of the inner product of the vector with itself).
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. 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 the type Real to each component of Left
and the matching component of Right. The index ranges of the result are
those of Left. 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. 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 operation "*"
of the type Real for computing the individual components. The first and
second index ranges of the 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). 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). 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 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 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 A'Range(2). 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 A'Range(2) and X'Range(2). 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) equal to the unit matrix. The index ranges of the result
are A'Range(2) and A'Range(1). 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 Eigenvalues(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. 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). Argument_Error is raised if the matrix A
is not symmetric.
procedure Eigensystem(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 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. Constraint_Error is raised
if A'Length(1) is not equal to A'Length(2), or if Values'Range is not
equal to A'Range(1), or if the index ranges of the parameter Vectors
are not equal to those of A. Argument_Error is raised if the matrix A
is not symmetric. Constraint_Error is also raised in implementation-defined
circumstances if the algorithm used does not converge quickly enough.
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.
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 * Real'Machine_Radix**(1 – Real'Model_Mantissa)
For the L2-norm, no accuracy requirements are specified
in the relaxed mode. In the strict mode the relative error on the norm
shall not exceed g / 2.0 + 3.0 * Real'Model_Epsilon where g
is defined as above.
Documentation Requirements
Implementations shall document any techniques used
to reduce cancellation errors such as extended precision arithmetic.
Implementation Permissions
The nongeneric equivalent packages can be actual
instantiations of the generic package for the appropriate predefined
type, though that is not required.
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 should be performed
by using the equality operator to compare the relevant components.
An implementation should minimize the circumstances
under which the algorithm used for Eigenvalues and Eigensystem fails
to converge.
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