diff options
Diffstat (limited to 'gcc/ada/libgnat/a-ngrear.adb')
| -rw-r--r-- | gcc/ada/libgnat/a-ngrear.adb | 85 |
1 files changed, 46 insertions, 39 deletions
diff --git a/gcc/ada/libgnat/a-ngrear.adb b/gcc/ada/libgnat/a-ngrear.adb index e7b1bcd..524b4a0 100644 --- a/gcc/ada/libgnat/a-ngrear.adb +++ b/gcc/ada/libgnat/a-ngrear.adb @@ -85,7 +85,7 @@ package body Ada.Numerics.Generic_Real_Arrays is function Is_Symmetric (A : Real_Matrix) return Boolean is (Transpose (A) = A); - -- Return True iff A is symmetric, see RM G.3.1 (90). + -- Return True iff A is symmetric, see RM G.3.1 (90) function Is_Tiny (Value, Compared_To : Real) return Boolean is (abs Compared_To + 100.0 * abs Value = abs Compared_To); @@ -104,7 +104,7 @@ package body Ada.Numerics.Generic_Real_Arrays is -- not a square matrix, and otherwise returns its length. procedure Rotate (X, Y : in out Real; Sin, Tau : Real); - -- Perform a Givens rotation + -- Perform a Givens rotation of angle Theta given by sin and sin/(1 + cos) procedure Sort_Eigensystem (Values : in out Real_Vector; @@ -525,13 +525,13 @@ package body Ada.Numerics.Generic_Real_Arrays is Vectors : out Real_Matrix; Compute_Vectors : Boolean) is - -- This subprogram uses Carl Gustav Jacob Jacobi's iterative method - -- for computing eigenvalues and eigenvectors and is based on + -- This subprogram uses Carl Gustav Jacob Jacobi's cyclic iterative + -- method for computing eigenvalues and eigenvectors and is based on -- Rutishauser's implementation. -- The given real symmetric matrix is transformed iteratively to -- diagonal form through a sequence of appropriately chosen elementary - -- orthogonal transformations, called Jacobi rotations here. + -- orthogonal transformations, called Jacobi rotations. -- The Jacobi method produces a systematic decrease of the sum of the -- squares of off-diagonal elements. Convergence to zero is quadratic, @@ -542,41 +542,44 @@ package body Ada.Numerics.Generic_Real_Arrays is -- best choice here, even though for large matrices other methods will -- be significantly more efficient in both time and space. - -- While the eigensystem computations are absolutely foolproof for all + -- While the eigensystem computation is absolutely foolproof for all -- real symmetric matrices, in presence of invalid values, or similar - -- exceptional situations it might not. In such cases the results cannot - -- be trusted and Constraint_Error is raised. + -- exceptional situations, it may not be. In such cases, the results + -- cannot be trusted and Constraint_Error is raised. -- Note: this implementation needs temporary storage for 2 * N + N**2 -- values of type Real. - Max_Iterations : constant := 50; - N : constant Natural := Length (A); + Max_Iterations : constant := 50; + N : constant Natural := Length (A); subtype Square_Matrix is Real_Matrix (1 .. N, 1 .. N); - -- In order to annihilate the M (Row, Col) element, the - -- rotation parameters Cos and Sin are computed as - -- follows: + -- In order to annihilate the M (Row, Col) element, the rotation angle + -- Theta is chosen as follows: - -- Theta = Cot (2.0 * Phi) - -- = (Diag (Col) - Diag (Row)) / (2.0 * M (Row, Col)) + -- Cot (2.0 * Theta) = (Diag (Col) - Diag (Row)) / (2.0 * M (Row, Col)) - -- Then Tan (Phi) as the smaller root (in modulus) of + -- If C = Cot (2.0 * Theta), then Tan (Theta) is computed as the smaller + -- root (in modulus) of: - -- T**2 + 2 * T * Theta = 1 (or 0.5 / Theta, if Theta is large) + -- X**2 + 2 * C * X - 1 = 0 - function Compute_Tan (Theta : Real) return Real is - (Real'Copy_Sign (1.0 / (abs Theta + Sqrt (1.0 + Theta**2)), Theta)); + -- or else as 0.5 / C, if C is large. + + function Compute_Tan (C : Real) return Real is + (Real'Copy_Sign (1.0 / (abs C + Sqrt (1.0 + C**2)), C)); function Compute_Tan (P, H : Real) return Real is - (if Is_Tiny (P, Compared_To => H) then P / H - else Compute_Tan (Theta => H / (2.0 * P))); + (if Is_Tiny (P, Compared_To => H) + then P / H + else Compute_Tan (C => H / (2.0 * P))); pragma Annotate (CodePeer, False_Positive, "divide by zero", "H, P /= 0"); function Sum_Strict_Upper (M : Square_Matrix) return Real; - -- Return the sum of all elements in the strict upper triangle of M + -- Return the sum of the absolute value of all the elements in the + -- strict upper triangle of M. ---------------------- -- Sum_Strict_Upper -- @@ -595,11 +598,13 @@ package body Ada.Numerics.Generic_Real_Arrays is return Sum; end Sum_Strict_Upper; + -- Local variables + M : Square_Matrix := A; -- Work space for solving eigensystem - Threshold : Real; - Sum : Real; Diag : Real_Vector (1 .. N); Diag_Adj : Real_Vector (1 .. N); + Sum : Real; + Threshold : Real; -- The vector Diag_Adj indicates the amount of change in each value, -- while Diag tracks the value itself and Values holds the values as @@ -621,22 +626,24 @@ package body Ada.Numerics.Generic_Real_Arrays is raise Constraint_Error with "matrix not symmetric"; end if; + Values := Diagonal (M); + -- Note: Only the locally declared matrix M and vectors (Diag, Diag_Adj) -- have lower bound equal to 1. The Vectors matrix may have -- different bounds, so take care indexing elements. Assignment -- as a whole is fine as sliding is automatic in that case. - Vectors := (if not Compute_Vectors then [1 .. 0 => [1 .. 0 => 0.0]] - else Unit_Matrix (Vectors'Length (1), Vectors'Length (2))); - Values := Diagonal (M); + Vectors := (if Compute_Vectors + then Unit_Matrix (N) + else [1 .. 0 => [1 .. 0 => 0.0]]); Sweep : for Iteration in 1 .. Max_Iterations loop - -- The first three iterations, perform rotation for any non-zero - -- element. After this, rotate only for those that are not much - -- smaller than the average off-diagnal element. After the fifth - -- iteration, additionally zero out off-diagonal elements that are - -- very small compared to elements on the diagonal with the same + -- During the first three iterations, perform the rotation only for + -- elements that are not much smaller than the average off-diagonal + -- element. After this, rotate for any non-zero elements. After the + -- fifth iteration, additionally zero out off-diagonal elements that + -- are very small compared to elements on the diagonal with the same -- column or row index. Sum := Sum_Strict_Upper (M); @@ -645,8 +652,8 @@ package body Ada.Numerics.Generic_Real_Arrays is Threshold := (if Iteration < 4 then 0.2 * Sum / Real (N**2) else 0.0); - -- Iterate over all off-diagonal elements, rotating any that have - -- an absolute value that exceeds the threshold. + -- Iterate over all off-diagonal elements, rotating any that have an + -- absolute value that exceeds the threshold. Diag := Values; Diag_Adj := [others => 0.0]; -- Accumulates adjustments to Diag @@ -654,11 +661,11 @@ package body Ada.Numerics.Generic_Real_Arrays is for Row in 1 .. N - 1 loop for Col in Row + 1 .. N loop - -- If, before the rotation M (Row, Col) is tiny compared to + -- If, before the rotation, M (Row, Col) is tiny compared to -- Diag (Row) and Diag (Col), rotation is skipped. This is -- meaningful, as it produces no larger error than would be -- produced anyhow if the rotation had been performed. - -- Suppress this optimization in the first four sweeps, so + -- Suppress this optimization in the first four iterations, so -- that this procedure can be used for computing eigenvectors -- of perturbed diagonal matrices. @@ -670,8 +677,8 @@ package body Ada.Numerics.Generic_Real_Arrays is elsif abs M (Row, Col) > Threshold then Perform_Rotation : declare - Tan : constant Real := Compute_Tan (M (Row, Col), - Diag (Col) - Diag (Row)); + Tan : constant Real := + Compute_Tan (M (Row, Col), Diag (Col) - Diag (Row)); Cos : constant Real := 1.0 / Sqrt (1.0 + Tan**2); Sin : constant Real := Tan * Cos; Tau : constant Real := Sin / (1.0 + Cos); @@ -710,7 +717,7 @@ package body Ada.Numerics.Generic_Real_Arrays is Values := Values + Diag_Adj; end loop Sweep; - -- All normal matrices with valid values should converge perfectly. + -- All normal matrices with valid values should converge perfectly if Sum /= 0.0 then raise Constraint_Error with "eigensystem solution does not converge"; |