G.3.2 Complex Vectors and Matrices
Static Semantics
{
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The generic library package Numerics.Generic_Complex_Arrays has the following
declaration:
{
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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
with Pure, Nonblocking
is -- 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;
{
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function "
abs" (Right : Complex_Vector)
return Real'Base;
-- 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;
-- Eigenvalues and vectors of a Hermitian matrix
function Eigenvalues(A : Complex_Matrix)
return Real_Vector;
procedure Eigensystem(A :
in Complex_Matrix;
Values :
out Real_Vector;
Vectors :
out Complex_Matrix);
-- Other Complex_Matrix operations
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Complex_Matrix;
end Ada.Numerics.Generic_Complex_Arrays;
{
AI95-00296-01}
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.
{
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Two types are defined and exported by 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.
{
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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
are as defined for the scalar operation.
{
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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.
Discussion: {
AI05-0047-1}
An inner product never involves implicit complex conjugation. If the
product of a vector with the conjugate of another (or the same) vector
is required, then this has to be stated explicitly by writing for example
X * Conjugate(Y). This mimics the usual mathematical notation.
function Re (X : Complex_Vector) return Real_Vector;
function Im (X : Complex_Vector) return Real_Vector;
{
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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);
{
AI95-00296-01}
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.
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;
{
AI95-00296-01}
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. 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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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.
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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. Constraint_Error
is raised if Left'Length is not equal to Right'Length.
function "*" (Left, Right : Complex_Vector) return Complex;
{
AI95-00296-01}
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.
{
AI05-0047-1}
function "
abs" (Right : Complex_Vector)
return Real'Base;
{
AI95-00418-01}
This operation returns the Hermitian L2-norm of Right (the square root
of the inner product of the vector with its conjugate).
Implementation Note: While the definition
is given in terms of an inner product, the norm doesn't “involve
an inner product” in the technical sense. The reason is that it
has accuracy requirements substantially different from those applicable
to inner products; and that cancellations cannot occur, because all the
terms are positive, so there is no possibility of intermediate overflow.
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;
{
AI95-00296-01}
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. 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;
{
AI95-00296-01}
Each operation returns the inner product of Left and Right. 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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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).
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;
{
AI95-00296-01}
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);
{
AI95-00296-01}
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.
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;
{
AI95-00296-01}
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. 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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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.
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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.
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;
{
AI95-00296-01}
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 : Complex_Vector) return Complex_Matrix;
{
AI95-00296-01}
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 result are Left'Range and Right'Range
respectively.
function "*" (Left : Complex_Vector;
Right : Complex_Matrix) return Complex_Vector;
{
AI95-00296-01}
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 : Complex_Matrix;
Right : Complex_Vector) return Complex_Vector;
{
AI95-00296-01}
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_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;
{
AI95-00296-01}
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.
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;
{
AI95-00296-01}
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. 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;
{
AI95-00296-01}
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 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;
{
AI95-00296-01}
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). 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;
{
AI95-00296-01}
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). 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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
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;
{
AI95-00296-01}
Each operation returns the result of applying the corresponding operation
in numerics.generic_complex_types to each component of the matrix Left
and the real number Right. The index ranges of the result are those of
Left.
function Solve (A : Complex_Matrix; X : Complex_Vector) return Complex_Vector;
{
AI95-00296-01}
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.
Discussion: The text says that Y is such
that “X is (nearly) equal to A * Y” rather than “X
is equal to A * Y” because rounding errors may mean that there
is no value of Y such that X is exactly equal to A * Y. On the other
hand it does not mean that any old rough value will do. The algorithm
given under Implementation Advice should be followed.
The requirement to raise Constraint_Error if
the matrix is ill-conditioned is really a reflection of what will happen
if the matrix is ill-conditioned. See Implementation Advice. We do not
make any attempt to define ill-conditioned formally.
These remarks apply to all versions of Solve
and Inverse.
function Solve (A, X : Complex_Matrix) return Complex_Matrix;
{
AI95-00296-01}
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 : Complex_Matrix) return Complex_Matrix;
{
AI95-00296-01}
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 : Complex_Matrix) return Complex;
{
AI95-00296-01}
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 : Complex_Matrix) return Real_Vector;
{
AI95-00296-01}
This function returns the eigenvalues of the Hermitian 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 Hermitian.
Discussion: A Hermitian matrix is one
whose transpose is equal to its complex conjugate. The eigenvalues of
a Hermitian matrix are always real. We only support this case because
algorithms for solving the general case are inherently unstable.
procedure Eigensystem(A : in Complex_Matrix;
Values : out Real_Vector;
Vectors : out Complex_Matrix);
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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.
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 Hermitian. Constraint_Error is also
raised in implementation-defined circumstances if the algorithm used
does not converge quickly enough.
Ramification: {
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There is no requirement on the absolute direction of the returned eigenvectors.
Thus they might be multiplied by any complex number whose modulus is
1. It is only the ratios of the components that matter. This is standard
practice.
function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Complex_Matrix;
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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). Constraint_Error is raised if First_1 + Order –
1 > Integer'Last or First_2 + Order – 1 > Integer'Last.
Implementation Requirements
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Accuracy requirements for the subprograms Solve, Inverse, Determinant,
Eigenvalues and Eigensystem are implementation defined.
Implementation defined: The accuracy
requirements for the subprograms Solve, Inverse, Determinant, Eigenvalues
and Eigensystem for type Complex_Matrix.
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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).
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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 mixed complex and real operands
g = sqrt(2.0) * X'Length * Real'Machine_Radix**(1 – Real'Model_Mantissa)
for two complex operands
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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 has the definition
appropriate for two complex operands.
Documentation Requirements
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Implementations shall document any techniques used to reduce cancellation
errors such as extended precision arithmetic.
Documentation Requirement: Any techniques
used to reduce cancellation errors in Numerics.Generic_Complex_Arrays
shall be documented.
Implementation 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
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The nongeneric equivalent packages can be actual instantiations of the
generic package for the appropriate predefined type, though that is not
required.
{
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{
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Although many operations are defined in terms of operations from Numerics.Generic_Complex_Types,
they can be implemented by other operations that have the same effect.
Implementation Advice
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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.
Implementation Advice: Solve and Inverse
for Numerics.Generic_Complex_Arrays should be implemented using established
techniques and the result should be refined by an iteration on the residuals.
{
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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.
Discussion: There isn't any advice for
the implementation to document with this paragraph.
{
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The test that a matrix is Hermitian should use the equality operator
to compare the real components and negation followed by equality to compare
the imaginary components (see
G.2.1).
Implementation Advice: The equality and
negation operators should be used to test that a matrix is Hermitian.
An implementation should minimize the circumstances
under which the algorithm used for Eigenvalues and Eigensystem fails
to converge.
Implementation Advice: An implementation
should minimize the circumstances under which the algorithm used for
Numerics.Generic_Complex_Arrays.Eigenvalues and Numerics.Generic_Complex_Arrays.Eigensystem
fails to converge.
Implementation Note: J. H. Wilkinson
is the acknowledged expert in this area. See for example Wilkinson, J.
H., and Reinsch, C. , Linear Algebra , vol II of Handbook for Automatic
Computation, Springer-Verlag, or Wilkinson, J. H., The Algebraic Eigenvalue
Problem, Oxford University Press.
{
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Implementations should not perform operations on mixed complex and real
operands by first converting the real operand to complex. See
G.1.1.
Implementation Advice: Mixed real and
complex operations should not be performed by converting the real operand
to complex.
Extensions to Ada 95
{
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The package Numerics.Generic_Complex_Arrays and
its nongeneric equivalents are new.
Wording Changes from Ada 2005
{
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Correction: Corrected various accuracy and definition issues.
Ada 2005 and 2012 Editions sponsored in part by Ada-Europe
