LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
DOUBLE PRECISION function  dlansy (NORM, UPLO, N, A, LDA, WORK) 
DLANSY returns the value of the 1norm, 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
DOUBLE PRECISION function dlansy  (  character  NORM, 
character  UPLO,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  WORK  
) 
DLANSY returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
Download DLANSY + dependencies [TGZ] [ZIP] [TXT]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.
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.
[in]  NORM  NORM is CHARACTER*1 Specifies the value to be returned in DLANSY as described above. 
[in]  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 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANSY is set to zero. 
[in]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(N,1). 
[out]  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. 
Definition at line 123 of file dlansy.f.
subroutine dlaqsy  (  character  UPLO, 
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  S,  
double precision  SCOND,  
double precision  AMAX,  
character  EQUED  
) 
DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Download DLAQSY + dependencies [TGZ] [ZIP] [TXT]DLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S.
[in]  UPLO  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 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  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). 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(N,1). 
[in]  S  S is DOUBLE PRECISION array, dimension (N) The scale factors for A. 
[in]  SCOND  SCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i). 
[in]  AMAX  AMAX is DOUBLE PRECISION Absolute value of largest matrix entry. 
[out]  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). 
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.
Definition at line 134 of file dlaqsy.f.
subroutine dlasy2  (  logical  LTRANL, 
logical  LTRANR,  
integer  ISGN,  
integer  N1,  
integer  N2,  
double precision, dimension( ldtl, * )  TL,  
integer  LDTL,  
double precision, dimension( ldtr, * )  TR,  
integer  LDTR,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision  SCALE,  
double precision, dimension( ldx, * )  X,  
integer  LDX,  
double precision  XNORM,  
integer  INFO  
) 
DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
Download DLASY2 + dependencies [TGZ] [ZIP] [TXT]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.
[in]  LTRANL  LTRANL is LOGICAL On entry, LTRANL specifies the op(TL): = .FALSE., op(TL) = TL, = .TRUE., op(TL) = TL**T. 
[in]  LTRANR  LTRANR is LOGICAL On entry, LTRANR specifies the op(TR): = .FALSE., op(TR) = TR, = .TRUE., op(TR) = TR**T. 
[in]  ISGN  ISGN is INTEGER On entry, ISGN specifies the sign of the equation as described before. ISGN may only be 1 or 1. 
[in]  N1  N1 is INTEGER On entry, N1 specifies the order of matrix TL. N1 may only be 0, 1 or 2. 
[in]  N2  N2 is INTEGER On entry, N2 specifies the order of matrix TR. N2 may only be 0, 1 or 2. 
[in]  TL  TL is DOUBLE PRECISION array, dimension (LDTL,2) On entry, TL contains an N1 by N1 matrix. 
[in]  LDTL  LDTL is INTEGER The leading dimension of the matrix TL. LDTL >= max(1,N1). 
[in]  TR  TR is DOUBLE PRECISION array, dimension (LDTR,2) On entry, TR contains an N2 by N2 matrix. 
[in]  LDTR  LDTR is INTEGER The leading dimension of the matrix TR. LDTR >= max(1,N2). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,2) On entry, the N1 by N2 matrix B contains the righthand side of the equation. 
[in]  LDB  LDB is INTEGER The leading dimension of the matrix B. LDB >= max(1,N1). 
[out]  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. 
[out]  X  X is DOUBLE PRECISION array, dimension (LDX,2) On exit, X contains the N1 by N2 solution. 
[in]  LDX  LDX is INTEGER The leading dimension of the matrix X. LDX >= max(1,N1). 
[out]  XNORM  XNORM is DOUBLE PRECISION On exit, XNORM is the infinitynorm of the solution. 
[out]  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. 
Definition at line 174 of file dlasy2.f.
subroutine dsyswapr  (  character  UPLO, 
integer  N,  
double precision, dimension( lda, n )  A,  
integer  LDA,  
integer  I1,  
integer  I2  
) 
DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
Download DSYSWAPR + dependencies [TGZ] [ZIP] [TXT]DSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix.
[in]  UPLO  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the NB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  I1  I1 is INTEGER Index of the first row to swap 
[in]  I2  I2 is INTEGER Index of the second row to swap 
Definition at line 103 of file dsyswapr.f.
subroutine dtgsy2  (  character  TRANS, 
integer  IJOB,  
integer  M,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( ldc, * )  C,  
integer  LDC,  
double precision, dimension( ldd, * )  D,  
integer  LDD,  
double precision, dimension( lde, * )  E,  
integer  LDE,  
double precision, dimension( ldf, * )  F,  
integer  LDF,  
double precision  SCALE,  
double precision  RDSUM,  
double precision  RDSCAL,  
integer, dimension( * )  IWORK,  
integer  PQ,  
integer  INFO  
) 
DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
Download DTGSY2 + dependencies [TGZ] [ZIP] [TXT]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 MbyN matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size MbyM, NbyN and MbyN, 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 communicaton 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 subpencils of the matrix pair in DTGSYL. See DTGSYL for details.
[in]  TRANS  TRANS is CHARACTER*1 = 'N', solve the generalized Sylvester equation (1). = 'T': solve the 'transposed' system (3). 
[in]  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 normbased estimate of the separation between two matrix pairs is computed. (look ahead strategy is used). = 2: A contribution from this subsystem to a Frobenius normbased estimate of the separation between two matrix pairs is computed. (DGECON on subsystems is used.) Not referenced if TRANS = 'T'. 
[in]  M  M is INTEGER On entry, M specifies the order of A and D, and the row dimension of C, F, R and L. 
[in]  N  N is INTEGER On entry, N specifies the order of B and E, and the column dimension of C, F, R and L. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, M) On entry, A contains an upper quasi triangular matrix. 
[in]  LDA  LDA is INTEGER The leading dimension of the matrix A. LDA >= max(1, M). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB, N) On entry, B contains an upper quasi triangular matrix. 
[in]  LDB  LDB is INTEGER The leading dimension of the matrix B. LDB >= max(1, N). 
[in,out]  C  C is DOUBLE PRECISION array, dimension (LDC, N) On entry, C contains the righthandside of the first matrix equation in (1). On exit, if IJOB = 0, C has been overwritten by the solution R. 
[in]  LDC  LDC is INTEGER The leading dimension of the matrix C. LDC >= max(1, M). 
[in]  D  D is DOUBLE PRECISION array, dimension (LDD, M) On entry, D contains an upper triangular matrix. 
[in]  LDD  LDD is INTEGER The leading dimension of the matrix D. LDD >= max(1, M). 
[in]  E  E is DOUBLE PRECISION array, dimension (LDE, N) On entry, E contains an upper triangular matrix. 
[in]  LDE  LDE is INTEGER The leading dimension of the matrix E. LDE >= max(1, N). 
[in,out]  F  F is DOUBLE PRECISION array, dimension (LDF, N) On entry, F contains the righthandside of the second matrix equation in (1). On exit, if IJOB = 0, F has been overwritten by the solution L. 
[in]  LDF  LDF is INTEGER The leading dimension of the matrix F. LDF >= max(1, M). 
[out]  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. 
[in,out]  RDSUM  RDSUM is DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Difestimate 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 subsystem. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL. 
[in,out]  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. 
[out]  IWORK  IWORK is INTEGER array, dimension (M+N+2) 
[out]  PQ  PQ is INTEGER On exit, the number of subsystems (of size 2by2, 4by4 and 8by8) solved by this routine. 
[out]  INFO  INFO is INTEGER On exit, if INFO is set to =0: Successful exit <0: If INFO = i, the ith argument had an illegal value. >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues. 
Definition at line 273 of file dtgsy2.f.