double precision function dlansy (NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
subroutine dlaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine dlasy2 (LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO)
DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
subroutine dsyswapr (UPLO, N, A, LDA, I1, I2)
DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
subroutine dtgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO)
DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
This is the group of double auxiliary functions for SY matrices
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
Purpose:
DLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A.
Returns
DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM is CHARACTER*1
Specifies the value to be returned in DLANSY as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is to be referenced.
= 'U': Upper triangular part of A is referenced
= 'L': Lower triangular part of A is referenced
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, DLANSY is
set to zero.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading n by n
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = 'L', the leading n by n lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(N,1).
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Purpose:
DLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S.
Parameters
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if EQUED = 'Y', the equilibrated matrix:
diag(S) * A * diag(S).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(N,1).
S
S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.
SCOND
SCOND is DOUBLE PRECISION
Ratio of the smallest S(i) to the largest S(i).
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix entry.
EQUED
EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
Internal Parameters:
THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
Purpose:
DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
op(TL)*X + ISGN*X*op(TR) = SCALE*B,
where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
-1. op(T) = T or T**T, where T**T denotes the transpose of T.
Parameters
LTRANL is LOGICAL
On entry, LTRANL specifies the op(TL):
= .FALSE., op(TL) = TL,
= .TRUE., op(TL) = TL**T.
LTRANR
LTRANR is LOGICAL
On entry, LTRANR specifies the op(TR):
= .FALSE., op(TR) = TR,
= .TRUE., op(TR) = TR**T.
ISGN
ISGN is INTEGER
On entry, ISGN specifies the sign of the equation
as described before. ISGN may only be 1 or -1.
N1
N1 is INTEGER
On entry, N1 specifies the order of matrix TL.
N1 may only be 0, 1 or 2.
N2
N2 is INTEGER
On entry, N2 specifies the order of matrix TR.
N2 may only be 0, 1 or 2.
TL
TL is DOUBLE PRECISION array, dimension (LDTL,2)
On entry, TL contains an N1 by N1 matrix.
LDTL
LDTL is INTEGER
The leading dimension of the matrix TL. LDTL >= max(1,N1).
TR
TR is DOUBLE PRECISION array, dimension (LDTR,2)
On entry, TR contains an N2 by N2 matrix.
LDTR
LDTR is INTEGER
The leading dimension of the matrix TR. LDTR >= max(1,N2).
B
B is DOUBLE PRECISION array, dimension (LDB,2)
On entry, the N1 by N2 matrix B contains the right-hand
side of the equation.
LDB
LDB is INTEGER
The leading dimension of the matrix B. LDB >= max(1,N1).
SCALE
SCALE is DOUBLE PRECISION
On exit, SCALE contains the scale factor. SCALE is chosen
less than or equal to 1 to prevent the solution overflowing.
X
X is DOUBLE PRECISION array, dimension (LDX,2)
On exit, X contains the N1 by N2 solution.
LDX
LDX is INTEGER
The leading dimension of the matrix X. LDX >= max(1,N1).
XNORM
XNORM is DOUBLE PRECISION
On exit, XNORM is the infinity-norm of the solution.
INFO
INFO is INTEGER
On exit, INFO is set to
0: successful exit.
1: TL and TR have too close eigenvalues, so TL or
TR is perturbed to get a nonsingular equation.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
Purpose:
DSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix.
Parameters
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA,*)
On entry, the N-by-N matrix A. On exit, the permuted matrix
where the rows I1 and I2 and columns I1 and I2 are interchanged.
If UPLO = 'U', the interchanges are applied to the upper
triangular part and the strictly lower triangular part of A is
not referenced; if UPLO = 'L', the interchanges are applied to
the lower triangular part and the part of A above the diagonal
is not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
I1
I1 is INTEGER
Index of the first row to swap
I2
I2 is INTEGER
Index of the second row to swap
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
Purpose:
DTGSY2 solves the generalized Sylvester equation:
A * R - L * B = scale * C (1)
D * R - L * E = scale * F,
using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
(A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
must be in generalized Schur canonical form, i.e. A, B are upper
quasi triangular and D, E are upper triangular. The solution (R, L)
overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
chosen to avoid overflow.
In matrix notation solving equation (1) corresponds to solve
Z*x = scale*b, where Z is defined as
Z = [ kron(In, A) -kron(B**T, Im) ] (2)
[ kron(In, D) -kron(E**T, Im) ],
Ik is the identity matrix of size k and X**T is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
In the process of solving (1), we solve a number of such systems
where Dim(In), Dim(In) = 1 or 2.
If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
which is equivalent to solve for R and L in
A**T * R + D**T * L = scale * C (3)
R * B**T + L * E**T = scale * -F
This case is used to compute an estimate of Dif[(A, D), (B, E)] =
sigma_min(Z) using reverse communication with DLACON.
DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
of an upper bound on the separation between to matrix pairs. Then
the input (A, D), (B, E) are sub-pencils of the matrix pair in
DTGSYL. See DTGSYL for details.
Parameters
TRANS is CHARACTER*1
= 'N': solve the generalized Sylvester equation (1).
= 'T': solve the 'transposed' system (3).
IJOB
IJOB is INTEGER
Specifies what kind of functionality to be performed.
= 0: solve (1) only.
= 1: A contribution from this subsystem to a Frobenius
norm-based estimate of the separation between two matrix
pairs is computed. (look ahead strategy is used).
= 2: A contribution from this subsystem to a Frobenius
norm-based estimate of the separation between two matrix
pairs is computed. (DGECON on sub-systems is used.)
Not referenced if TRANS = 'T'.
M
M is INTEGER
On entry, M specifies the order of A and D, and the row
dimension of C, F, R and L.
N
N is INTEGER
On entry, N specifies the order of B and E, and the column
dimension of C, F, R and L.
A
A is DOUBLE PRECISION array, dimension (LDA, M)
On entry, A contains an upper quasi triangular matrix.
LDA
LDA is INTEGER
The leading dimension of the matrix A. LDA >= max(1, M).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, B contains an upper quasi triangular matrix.
LDB
LDB is INTEGER
The leading dimension of the matrix B. LDB >= max(1, N).
C
C is DOUBLE PRECISION array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1).
On exit, if IJOB = 0, C has been overwritten by the
solution R.
LDC
LDC is INTEGER
The leading dimension of the matrix C. LDC >= max(1, M).
D
D is DOUBLE PRECISION array, dimension (LDD, M)
On entry, D contains an upper triangular matrix.
LDD
LDD is INTEGER
The leading dimension of the matrix D. LDD >= max(1, M).
E
E is DOUBLE PRECISION array, dimension (LDE, N)
On entry, E contains an upper triangular matrix.
LDE
LDE is INTEGER
The leading dimension of the matrix E. LDE >= max(1, N).
F
F is DOUBLE PRECISION array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1).
On exit, if IJOB = 0, F has been overwritten by the
solution L.
LDF
LDF is INTEGER
The leading dimension of the matrix F. LDF >= max(1, M).
SCALE
SCALE is DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
R and L (C and F on entry) will hold the solutions to a
slightly perturbed system but the input matrices A, B, D and
E have not been changed. If SCALE = 0, R and L will hold the
solutions to the homogeneous system with C = F = 0. Normally,
SCALE = 1.
RDSUM
RDSUM is DOUBLE PRECISION
On entry, the sum of squares of computed contributions to
the Dif-estimate under computation by DTGSYL, where the
scaling factor RDSCAL (see below) has been factored out.
On exit, the corresponding sum of squares updated with the
contributions from the current sub-system.
If TRANS = 'T' RDSUM is not touched.
NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.
RDSCAL
RDSCAL is DOUBLE PRECISION
On entry, scaling factor used to prevent overflow in RDSUM.
On exit, RDSCAL is updated w.r.t. the current contributions
in RDSUM.
If TRANS = 'T', RDSCAL is not touched.
NOTE: RDSCAL only makes sense when DTGSY2 is called by
DTGSYL.
IWORK
IWORK is INTEGER array, dimension (M+N+2)
PQ
PQ is INTEGER
On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
8-by-8) solved by this routine.
INFO
INFO is INTEGER
On exit, if INFO is set to
=0: Successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: The matrix pairs (A, D) and (B, E) have common or very
close eigenvalues.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
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