| ------------------------------------------------------------------------------ |
| -- -- |
| -- GNAT RUN-TIME COMPONENTS -- |
| -- -- |
| -- S Y S T E M . G E N E R I C _ A R R A Y _ O P E R A T I O N S -- |
| -- -- |
| -- S p e c -- |
| -- -- |
| -- Copyright (C) 2006-2022, Free Software Foundation, Inc. -- |
| -- -- |
| -- GNAT is free software; you can redistribute it and/or modify it under -- |
| -- terms of the GNU General Public License as published by the Free Soft- -- |
| -- ware Foundation; either version 3, or (at your option) any later ver- -- |
| -- sion. GNAT is distributed in the hope that it will be useful, but WITH- -- |
| -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -- |
| -- or FITNESS FOR A PARTICULAR PURPOSE. -- |
| -- -- |
| -- As a special exception under Section 7 of GPL version 3, you are granted -- |
| -- additional permissions described in the GCC Runtime Library Exception, -- |
| -- version 3.1, as published by the Free Software Foundation. -- |
| -- -- |
| -- You should have received a copy of the GNU General Public License and -- |
| -- a copy of the GCC Runtime Library Exception along with this program; -- |
| -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see -- |
| -- <http://www.gnu.org/licenses/>. -- |
| -- -- |
| -- GNAT was originally developed by the GNAT team at New York University. -- |
| -- Extensive contributions were provided by Ada Core Technologies Inc. -- |
| -- -- |
| ------------------------------------------------------------------------------ |
| |
| -- Proof of this unit is only done up to silver level, i.e. absence of runtime |
| -- errors, and only regarding runtime checks that depend on the generic part, |
| -- ignoring runtime checks related to formal generic subprogram parameters |
| -- in instantiations. For example, contracts do not protect against scalar |
| -- overflows in arithmetic operations passed on as formal generic subprogram |
| -- parameters. |
| |
| -- Preconditions in this unit are meant for analysis only, not for run-time |
| -- checking, so that the expected exceptions are raised. This is enforced |
| -- by setting the corresponding assertion policy to Ignore. Postconditions |
| -- and contract cases should not be executed at runtime as well, in order |
| -- not to slow down the execution of these functions. |
| |
| pragma Assertion_Policy (Pre => Ignore, |
| Post => Ignore, |
| Contract_Cases => Ignore, |
| Ghost => Ignore); |
| |
| package System.Generic_Array_Operations |
| with SPARK_Mode |
| is |
| |
| pragma Pure (Generic_Array_Operations); |
| |
| --------------------- |
| -- Back_Substitute -- |
| --------------------- |
| |
| generic |
| type Scalar is private; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| with function "-" (Left, Right : Scalar) return Scalar is <>; |
| with function "*" (Left, Right : Scalar) return Scalar is <>; |
| with function "/" (Left, Right : Scalar) return Scalar is <>; |
| with function Is_Non_Zero (X : Scalar) return Boolean is <>; |
| procedure Back_Substitute (M, N : in out Matrix) |
| with |
| Pre => M'First (1) = N'First (1) |
| and then M'Last (1) = N'Last (1); |
| |
| -------------- |
| -- Diagonal -- |
| -------------- |
| |
| generic |
| type Scalar is private; |
| type Vector is array (Integer range <>) of Scalar; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| function Diagonal (A : Matrix) return Vector |
| with |
| Pre => A'First (1) < A'Last (1) |
| and then A'First (2) < A'Last (2) |
| and then (if A'First (1) <= 0 then |
| A'Last (1) < Integer'Last + A'First (1)) |
| and then (if A'First (2) <= 0 then |
| A'Last (2) < Integer'Last + A'First (2)); |
| |
| ----------------------- |
| -- Forward_Eliminate -- |
| ----------------------- |
| |
| -- Use elementary row operations to put square matrix M in row echolon |
| -- form. Identical row operations are performed on matrix N, must have the |
| -- same number of rows as M. |
| |
| generic |
| type Scalar is private; |
| type Real is digits <>; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| with function "abs" (Right : Scalar) return Real'Base is <>; |
| with function "-" (Left, Right : Scalar) return Scalar is <>; |
| with function "*" (Left, Right : Scalar) return Scalar is <>; |
| with function "/" (Left, Right : Scalar) return Scalar is <>; |
| Zero : Scalar; |
| One : Scalar; |
| procedure Forward_Eliminate |
| (M : in out Matrix; |
| N : in out Matrix; |
| Det : out Scalar) |
| with |
| Pre => M'First (1) = N'First (1) |
| and then M'Last (1) = N'Last (1); |
| |
| -------------------------- |
| -- Square_Matrix_Length -- |
| -------------------------- |
| |
| generic |
| type Scalar is private; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| function Square_Matrix_Length (A : Matrix) return Natural |
| with |
| Pre => (if A'First (1) <= 0 then |
| A'Last (1) < Integer'Last + A'First (1)) |
| and then (if A'First (2) <= 0 then |
| A'Last (2) < Integer'Last + A'First (2)) |
| and then A'Length (1) = A'Length (2); |
| -- If A is non-square, raise Constraint_Error, else return its dimension |
| |
| ---------------------------------- |
| -- Vector_Elementwise_Operation -- |
| ---------------------------------- |
| |
| generic |
| type X_Scalar is private; |
| type Result_Scalar is private; |
| type X_Vector is array (Integer range <>) of X_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| with function Operation (X : X_Scalar) return Result_Scalar; |
| function Vector_Elementwise_Operation (X : X_Vector) return Result_Vector; |
| |
| ---------------------------------- |
| -- Matrix_Elementwise_Operation -- |
| ---------------------------------- |
| |
| generic |
| type X_Scalar is private; |
| type Result_Scalar is private; |
| type X_Matrix is array (Integer range <>, Integer range <>) of X_Scalar; |
| type Result_Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| with function Operation (X : X_Scalar) return Result_Scalar; |
| function Matrix_Elementwise_Operation (X : X_Matrix) return Result_Matrix; |
| |
| ----------------------------------------- |
| -- Vector_Vector_Elementwise_Operation -- |
| ----------------------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Vector is array (Integer range <>) of Left_Scalar; |
| type Right_Vector is array (Integer range <>) of Right_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| with function Operation |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar; |
| function Vector_Vector_Elementwise_Operation |
| (Left : Left_Vector; |
| Right : Right_Vector) return Result_Vector |
| with |
| Pre => (if Left'First <= 0 then |
| Left'Last < Integer'Last + Left'First) |
| and then (if Right'First <= 0 then |
| Right'Last < Integer'Last + Right'First) |
| and then Left'Length = Right'Length; |
| |
| ------------------------------------------------ |
| -- Vector_Vector_Scalar_Elementwise_Operation -- |
| ------------------------------------------------ |
| |
| generic |
| type X_Scalar is private; |
| type Y_Scalar is private; |
| type Z_Scalar is private; |
| type Result_Scalar is private; |
| type X_Vector is array (Integer range <>) of X_Scalar; |
| type Y_Vector is array (Integer range <>) of Y_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| with function Operation |
| (X : X_Scalar; |
| Y : Y_Scalar; |
| Z : Z_Scalar) return Result_Scalar; |
| function Vector_Vector_Scalar_Elementwise_Operation |
| (X : X_Vector; |
| Y : Y_Vector; |
| Z : Z_Scalar) return Result_Vector |
| with |
| Pre => (if X'First <= 0 then X'Last < Integer'Last + X'First) |
| and then (if Y'First <= 0 then Y'Last < Integer'Last + Y'First) |
| and then X'Length = Y'Length; |
| |
| ----------------------------------------- |
| -- Matrix_Matrix_Elementwise_Operation -- |
| ----------------------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Matrix is array (Integer range <>, Integer range <>) |
| of Left_Scalar; |
| type Right_Matrix is array (Integer range <>, Integer range <>) |
| of Right_Scalar; |
| type Result_Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| with function Operation |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar; |
| function Matrix_Matrix_Elementwise_Operation |
| (Left : Left_Matrix; |
| Right : Right_Matrix) return Result_Matrix |
| with |
| Pre => (if Left'First (1) <= 0 then |
| Left'Last (1) < Integer'Last + Left'First (1)) |
| and then (if Right'First (1) <= 0 then |
| Right'Last (1) < Integer'Last + Right'First (1)) |
| and then Left'Length (1) = Right'Length (1) |
| and then (if Left'First (2) <= 0 then |
| Left'Last (2) < Integer'Last + Left'First (2)) |
| and then (if Right'First (2) <= 0 then |
| Right'Last (2) < Integer'Last + Right'First (2)) |
| and then Left'Length (2) = Right'Length (2); |
| |
| ------------------------------------------------ |
| -- Matrix_Matrix_Scalar_Elementwise_Operation -- |
| ------------------------------------------------ |
| |
| generic |
| type X_Scalar is private; |
| type Y_Scalar is private; |
| type Z_Scalar is private; |
| type Result_Scalar is private; |
| type X_Matrix is array (Integer range <>, Integer range <>) of X_Scalar; |
| type Y_Matrix is array (Integer range <>, Integer range <>) of Y_Scalar; |
| type Result_Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| with function Operation |
| (X : X_Scalar; |
| Y : Y_Scalar; |
| Z : Z_Scalar) return Result_Scalar; |
| function Matrix_Matrix_Scalar_Elementwise_Operation |
| (X : X_Matrix; |
| Y : Y_Matrix; |
| Z : Z_Scalar) return Result_Matrix |
| with |
| Pre => (if X'First (1) <= 0 then |
| X'Last (1) < Integer'Last + X'First (1)) |
| and then (if Y'First (1) <= 0 then |
| Y'Last (1) < Integer'Last + Y'First (1)) |
| and then X'Length (1) = Y'Length (1) |
| and then (if X'First (2) <= 0 then |
| X'Last (2) < Integer'Last + X'First (2)) |
| and then (if Y'First (2) <= 0 then |
| Y'Last (2) < Integer'Last + Y'First (2)) |
| and then X'Length (2) = Y'Length (2); |
| |
| ----------------------------------------- |
| -- Vector_Scalar_Elementwise_Operation -- |
| ----------------------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Vector is array (Integer range <>) of Left_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| with function Operation |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar; |
| function Vector_Scalar_Elementwise_Operation |
| (Left : Left_Vector; |
| Right : Right_Scalar) return Result_Vector; |
| |
| ----------------------------------------- |
| -- Matrix_Scalar_Elementwise_Operation -- |
| ----------------------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Matrix is array (Integer range <>, Integer range <>) |
| of Left_Scalar; |
| type Result_Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| with function Operation |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar; |
| function Matrix_Scalar_Elementwise_Operation |
| (Left : Left_Matrix; |
| Right : Right_Scalar) return Result_Matrix; |
| |
| ----------------------------------------- |
| -- Scalar_Vector_Elementwise_Operation -- |
| ----------------------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Right_Vector is array (Integer range <>) of Right_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| with function Operation |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar; |
| function Scalar_Vector_Elementwise_Operation |
| (Left : Left_Scalar; |
| Right : Right_Vector) return Result_Vector; |
| |
| ----------------------------------------- |
| -- Scalar_Matrix_Elementwise_Operation -- |
| ----------------------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Right_Matrix is array (Integer range <>, Integer range <>) |
| of Right_Scalar; |
| type Result_Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| with function Operation |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar; |
| function Scalar_Matrix_Elementwise_Operation |
| (Left : Left_Scalar; |
| Right : Right_Matrix) return Result_Matrix; |
| |
| ------------------- |
| -- Inner_Product -- |
| ------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Vector is array (Integer range <>) of Left_Scalar; |
| type Right_Vector is array (Integer range <>) of Right_Scalar; |
| Zero : Result_Scalar; |
| with function "*" |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar is <>; |
| with function "+" |
| (Left : Result_Scalar; |
| Right : Result_Scalar) return Result_Scalar is <>; |
| function Inner_Product |
| (Left : Left_Vector; |
| Right : Right_Vector) return Result_Scalar |
| with |
| Pre => (if Left'First <= 0 then |
| Left'Last < Integer'Last + Left'First) |
| and then (if Right'First <= 0 then |
| Right'Last < Integer'Last + Right'First) |
| and then Left'Length = Right'Length; |
| |
| ------------- |
| -- L2_Norm -- |
| ------------- |
| |
| generic |
| type X_Scalar is private; |
| type Result_Real is digits <>; |
| type X_Vector is array (Integer range <>) of X_Scalar; |
| with function "abs" (Right : X_Scalar) return Result_Real is <>; |
| with function Sqrt (X : Result_Real'Base) return Result_Real'Base is <>; |
| function L2_Norm (X : X_Vector) return Result_Real'Base; |
| |
| ------------------- |
| -- Outer_Product -- |
| ------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Vector is array (Integer range <>) of Left_Scalar; |
| type Right_Vector is array (Integer range <>) of Right_Scalar; |
| type Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| with function "*" |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar is <>; |
| function Outer_Product |
| (Left : Left_Vector; |
| Right : Right_Vector) return Matrix; |
| |
| --------------------------- |
| -- Matrix_Vector_Product -- |
| --------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Matrix is array (Integer range <>, Integer range <>) |
| of Left_Scalar; |
| type Right_Vector is array (Integer range <>) of Right_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| Zero : Result_Scalar; |
| with function "*" |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar is <>; |
| with function "+" |
| (Left : Result_Scalar; |
| Right : Result_Scalar) return Result_Scalar is <>; |
| function Matrix_Vector_Product |
| (Left : Matrix; |
| Right : Right_Vector) return Result_Vector |
| with |
| Pre => (if Left'First (2) <= 0 then |
| Left'Last (2) < Integer'Last + Left'First (2)) |
| and then (if Right'First <= 0 then |
| Right'Last < Integer'Last + Right'First) |
| and then Left'Length (2) = Right'Length; |
| |
| --------------------------- |
| -- Vector_Matrix_Product -- |
| --------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Vector is array (Integer range <>) of Left_Scalar; |
| type Matrix is array (Integer range <>, Integer range <>) |
| of Right_Scalar; |
| type Result_Vector is array (Integer range <>) of Result_Scalar; |
| Zero : Result_Scalar; |
| with function "*" |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar is <>; |
| with function "+" |
| (Left : Result_Scalar; |
| Right : Result_Scalar) return Result_Scalar is <>; |
| function Vector_Matrix_Product |
| (Left : Left_Vector; |
| Right : Matrix) return Result_Vector |
| with |
| Pre => (if Left'First <= 0 then |
| Left'Last < Integer'Last + Left'First) |
| and then (if Right'First (1) <= 0 then |
| Right'Last (1) < Integer'Last + Right'First (1)) |
| and then Left'Length = Right'Length (1); |
| |
| --------------------------- |
| -- Matrix_Matrix_Product -- |
| --------------------------- |
| |
| generic |
| type Left_Scalar is private; |
| type Right_Scalar is private; |
| type Result_Scalar is private; |
| type Left_Matrix is array (Integer range <>, Integer range <>) |
| of Left_Scalar; |
| type Right_Matrix is array (Integer range <>, Integer range <>) |
| of Right_Scalar; |
| type Result_Matrix is array (Integer range <>, Integer range <>) |
| of Result_Scalar; |
| Zero : Result_Scalar; |
| with function "*" |
| (Left : Left_Scalar; |
| Right : Right_Scalar) return Result_Scalar is <>; |
| with function "+" |
| (Left : Result_Scalar; |
| Right : Result_Scalar) return Result_Scalar is <>; |
| function Matrix_Matrix_Product |
| (Left : Left_Matrix; |
| Right : Right_Matrix) return Result_Matrix |
| with |
| Pre => (if Left'First (2) <= 0 then |
| Left'Last (2) < Integer'Last + Left'First (2)) |
| and then (if Right'First (1) <= 0 then |
| Right'Last (1) < Integer'Last + Right'First (1)) |
| and then Left'Length (2) = Right'Length (1); |
| |
| ---------------------------- |
| -- Matrix_Vector_Solution -- |
| ---------------------------- |
| |
| generic |
| type Scalar is private; |
| Zero : Scalar; |
| type Vector is array (Integer range <>) of Scalar; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| with procedure Back_Substitute (M, N : in out Matrix) is <>; |
| with procedure Forward_Eliminate |
| (M : in out Matrix; |
| N : in out Matrix; |
| Det : out Scalar) is <>; |
| function Matrix_Vector_Solution (A : Matrix; X : Vector) return Vector |
| with |
| Pre => (if A'First (1) <= 0 then |
| A'Last (1) < Integer'Last + A'First (1)) |
| and then (if A'First (2) <= 0 then |
| A'Last (2) < Integer'Last + A'First (2)) |
| and then A'Length (1) = A'Length (2) |
| and then (if X'First <= 0 then |
| X'Last < Integer'Last + X'First) |
| and then A'Length (1) = X'Length; |
| |
| ---------------------------- |
| -- Matrix_Matrix_Solution -- |
| ---------------------------- |
| |
| generic |
| type Scalar is private; |
| Zero : Scalar; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| with procedure Back_Substitute (M, N : in out Matrix) is <>; |
| with procedure Forward_Eliminate |
| (M : in out Matrix; |
| N : in out Matrix; |
| Det : out Scalar) is <>; |
| function Matrix_Matrix_Solution (A : Matrix; X : Matrix) return Matrix |
| with |
| Pre => (if A'First (1) <= 0 then |
| A'Last (1) < Integer'Last + A'First (1)) |
| and then (if A'First (2) <= 0 then |
| A'Last (2) < Integer'Last + A'First (2)) |
| and then A'Length (1) = A'Length (2) |
| and then (if X'First (1) <= 0 then |
| X'Last (1) < Integer'Last + X'First (1)) |
| and then A'Length (1) = X'Length (1); |
| |
| ---------- |
| -- Sqrt -- |
| ---------- |
| |
| generic |
| type Real is digits <>; |
| function Sqrt (X : Real'Base) return Real'Base; |
| |
| ----------------- |
| -- Swap_Column -- |
| ----------------- |
| |
| generic |
| type Scalar is private; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| procedure Swap_Column (A : in out Matrix; Left, Right : Integer) |
| with |
| Pre => Left in A'Range (2) |
| and then Right in A'Range (2); |
| |
| --------------- |
| -- Transpose -- |
| --------------- |
| |
| generic |
| type Scalar is private; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| procedure Transpose (A : Matrix; R : out Matrix) |
| with |
| Relaxed_Initialization => R, |
| Pre => A'First (1) = R'First (2) |
| and then A'Last (1) = R'Last (2) |
| and then A'First (2) = R'First (1) |
| and then A'Last (2) = R'Last (1) |
| and then (if A'First (1) < 0 then |
| A'Last (1) <= Integer'Last + A'First (1)) |
| and then (if A'First (2) < 0 then |
| A'Last (2) <= Integer'Last + A'First (2)), |
| Post => R'Initialized; |
| |
| ------------------------------- |
| -- Update_Vector_With_Vector -- |
| ------------------------------- |
| |
| generic |
| type X_Scalar is private; |
| type Y_Scalar is private; |
| type X_Vector is array (Integer range <>) of X_Scalar; |
| type Y_Vector is array (Integer range <>) of Y_Scalar; |
| with procedure Update (X : in out X_Scalar; Y : Y_Scalar); |
| procedure Update_Vector_With_Vector (X : in out X_Vector; Y : Y_Vector) |
| with |
| Pre => (if X'First <= 0 then |
| X'Last < Integer'Last + X'First) |
| and then (if Y'First <= 0 then |
| Y'Last < Integer'Last + Y'First) |
| and then X'Length = Y'Length; |
| |
| ------------------------------- |
| -- Update_Matrix_With_Matrix -- |
| ------------------------------- |
| |
| generic |
| type X_Scalar is private; |
| type Y_Scalar is private; |
| type X_Matrix is array (Integer range <>, Integer range <>) of X_Scalar; |
| type Y_Matrix is array (Integer range <>, Integer range <>) of Y_Scalar; |
| with procedure Update (X : in out X_Scalar; Y : Y_Scalar); |
| procedure Update_Matrix_With_Matrix (X : in out X_Matrix; Y : Y_Matrix) |
| with |
| Pre => (if X'First (1) <= 0 then |
| X'Last (1) < Integer'Last + X'First (1)) |
| and then (if Y'First (1) <= 0 then |
| Y'Last (1) < Integer'Last + Y'First (1)) |
| and then X'Length (1) = Y'Length (1) |
| and then (if X'First (2) <= 0 then |
| X'Last (2) < Integer'Last + X'First (2)) |
| and then (if Y'First (2) <= 0 then |
| Y'Last (2) < Integer'Last + Y'First (2)) |
| and then X'Length (2) = Y'Length (2); |
| |
| ----------------- |
| -- Unit_Matrix -- |
| ----------------- |
| |
| generic |
| type Scalar is private; |
| type Matrix is array (Integer range <>, Integer range <>) of Scalar; |
| Zero : Scalar; |
| One : Scalar; |
| function Unit_Matrix |
| (Order : Positive; |
| First_1 : Integer := 1; |
| First_2 : Integer := 1) return Matrix |
| with |
| Pre => First_1 <= Integer'Last - Order + 1 |
| and then First_2 <= Integer'Last - Order + 1; |
| |
| ----------------- |
| -- Unit_Vector -- |
| ----------------- |
| |
| generic |
| type Scalar is private; |
| type Vector is array (Integer range <>) of Scalar; |
| Zero : Scalar; |
| One : Scalar; |
| function Unit_Vector |
| (Index : Integer; |
| Order : Positive; |
| First : Integer := 1) return Vector |
| with |
| Pre => Index >= First |
| and then First <= Integer'Last - Order + 1 |
| and then Index <= First + (Order - 1); |
| |
| end System.Generic_Array_Operations; |
