LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
subroutine  zlaesy (A, B, C, RT1, RT2, EVSCAL, CS1, SN1) 
ZLAESY computes the eigenvalues and eigenvectors of a 2by2 complex symmetric matrix.  
DOUBLE PRECISION function  zlansy (NORM, UPLO, N, A, LDA, WORK) 
ZLANSY returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.  
subroutine  zlaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED) 
ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.  
subroutine  zsymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) 
ZSYMV computes a matrixvector product for a complex symmetric matrix.  
subroutine  zsyr (UPLO, N, ALPHA, X, INCX, A, LDA) 
ZSYR performs the symmetric rank1 update of a complex symmetric matrix.  
subroutine  zsyswapr (UPLO, N, A, LDA, I1, I2) 
ZSYSWAPR  
subroutine  ztgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO) 
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm). 
This is the group of complex16 auxiliary functions for SY matrices
subroutine zlaesy  (  complex*16  A, 
complex*16  B,  
complex*16  C,  
complex*16  RT1,  
complex*16  RT2,  
complex*16  EVSCAL,  
complex*16  CS1,  
complex*16  SN1  
) 
ZLAESY computes the eigenvalues and eigenvectors of a 2by2 complex symmetric matrix.
Download ZLAESY + dependencies [TGZ] [ZIP] [TXT]ZLAESY computes the eigendecomposition of a 2by2 symmetric matrix ( ( A, B );( B, C ) ) provided the norm of the matrix of eigenvectors is larger than some threshold value. RT1 is the eigenvalue of larger absolute value, and RT2 of smaller absolute value. If the eigenvectors are computed, then on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence [ CS1 SN1 ] . [ A B ] . [ CS1 SN1 ] = [ RT1 0 ] [ SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ]
[in]  A  A is COMPLEX*16 The ( 1, 1 ) element of input matrix. 
[in]  B  B is COMPLEX*16 The ( 1, 2 ) element of input matrix. The ( 2, 1 ) element is also given by B, since the 2by2 matrix is symmetric. 
[in]  C  C is COMPLEX*16 The ( 2, 2 ) element of input matrix. 
[out]  RT1  RT1 is COMPLEX*16 The eigenvalue of larger modulus. 
[out]  RT2  RT2 is COMPLEX*16 The eigenvalue of smaller modulus. 
[out]  EVSCAL  EVSCAL is COMPLEX*16 The complex value by which the eigenvector matrix was scaled to make it orthonormal. If EVSCAL is zero, the eigenvectors were not computed. This means one of two things: the 2by2 matrix could not be diagonalized, or the norm of the matrix of eigenvectors before scaling was larger than the threshold value THRESH (set below). 
[out]  CS1  CS1 is COMPLEX*16 
[out]  SN1  SN1 is COMPLEX*16 If EVSCAL .NE. 0, ( CS1, SN1 ) is the unit right eigenvector for RT1. 
DOUBLE PRECISION function zlansy  (  character  NORM, 
character  UPLO,  
integer  N,  
complex*16, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  WORK  
) 
ZLANSY returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.
Download ZLANSY + dependencies [TGZ] [ZIP] [TXT]ZLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A.
ZLANSY = ( 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 ZLANSY 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, ZLANSY is set to zero. 
[in]  A  A is COMPLEX*16 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 124 of file zlansy.f.
subroutine zlaqsy  (  character  UPLO, 
integer  N,  
complex*16, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  S,  
double precision  SCOND,  
double precision  AMAX,  
character  EQUED  
) 
ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Download ZLAQSY + dependencies [TGZ] [ZIP] [TXT]ZLAQSY 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 COMPLEX*16 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 135 of file zlaqsy.f.
subroutine zsymv  (  character  UPLO, 
integer  N,  
complex*16  ALPHA,  
complex*16, dimension( lda, * )  A,  
integer  LDA,  
complex*16, dimension( * )  X,  
integer  INCX,  
complex*16  BETA,  
complex*16, dimension( * )  Y,  
integer  INCY  
) 
ZSYMV computes a matrixvector product for a complex symmetric matrix.
Download ZSYMV + dependencies [TGZ] [ZIP] [TXT]ZSYMV performs the matrixvector operation y := alpha*A*x + beta*y, where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix.
[in]  UPLO  UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. Unchanged on exit. 
[in]  N  N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit. 
[in]  ALPHA  ALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit. 
[in]  A  A is COMPLEX*16 array, dimension ( LDA, N ) Before entry, with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced. Before entry, with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced. Unchanged on exit. 
[in]  LDA  LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, N ). Unchanged on exit. 
[in]  X  X is COMPLEX*16 array, dimension at least ( 1 + ( N  1 )*abs( INCX ) ). Before entry, the incremented array X must contain the N element vector x. Unchanged on exit. 
[in]  INCX  INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. 
[in]  BETA  BETA is COMPLEX*16 On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit. 
[in,out]  Y  Y is COMPLEX*16 array, dimension at least ( 1 + ( N  1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y. 
[in]  INCY  INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit. 
Definition at line 158 of file zsymv.f.
subroutine zsyr  (  character  UPLO, 
integer  N,  
complex*16  ALPHA,  
complex*16, dimension( * )  X,  
integer  INCX,  
complex*16, dimension( lda, * )  A,  
integer  LDA  
) 
ZSYR performs the symmetric rank1 update of a complex symmetric matrix.
Download ZSYR + dependencies [TGZ] [ZIP] [TXT]ZSYR performs the symmetric rank 1 operation A := alpha*x*x**H + A, where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix.
[in]  UPLO  UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. Unchanged on exit. 
[in]  N  N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit. 
[in]  ALPHA  ALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit. 
[in]  X  X is COMPLEX*16 array, dimension at least ( 1 + ( N  1 )*abs( INCX ) ). Before entry, the incremented array X must contain the N element vector x. Unchanged on exit. 
[in]  INCX  INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. 
[in,out]  A  A is COMPLEX*16 array, dimension ( LDA, N ) Before entry, with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced. On exit, the upper triangular part of the array A is overwritten by the upper triangular part of the updated matrix. Before entry, with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced. On exit, the lower triangular part of the array A is overwritten by the lower triangular part of the updated matrix. 
[in]  LDA  LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, N ). Unchanged on exit. 
Definition at line 136 of file zsyr.f.
subroutine zsyswapr  (  character  UPLO, 
integer  N,  
complex*16, dimension( lda, n )  A,  
integer  LDA,  
integer  I1,  
integer  I2  
) 
ZSYSWAPR
Download ZSYSWAPR + dependencies [TGZ] [ZIP] [TXT]ZSYSWAPR 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 COMPLEX*16 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 ZSYTRF. 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 zsyswapr.f.
subroutine ztgsy2  (  character  TRANS, 
integer  IJOB,  
integer  M,  
integer  N,  
complex*16, dimension( lda, * )  A,  
integer  LDA,  
complex*16, dimension( ldb, * )  B,  
integer  LDB,  
complex*16, dimension( ldc, * )  C,  
integer  LDC,  
complex*16, dimension( ldd, * )  D,  
integer  LDD,  
complex*16, dimension( lde, * )  E,  
integer  LDE,  
complex*16, dimension( ldf, * )  F,  
integer  LDF,  
double precision  SCALE,  
double precision  RDSUM,  
double precision  RDSCAL,  
integer  INFO  
) 
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
Download ZTGSY2 + dependencies [TGZ] [ZIP] [TXT]ZTGSY2 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. A, B, D and E are upper triangular (i.e., (A,D) and (B,E) in generalized Schur form). 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 Zx = scale * b, where Z is defined as Z = [ kron(In, A) kron(B**H, Im) ] (2) [ kron(In, D) kron(E**H, Im) ], Ik is the identity matrix of size k and X**H is the conjuguate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b is solved for, which is equivalent to solve for R and L in A**H * R + D**H * L = scale * C (3) R * B**H + L * E**H = scale * F This case is used to compute an estimate of Dif[(A, D), (B, E)] = = sigma_min(Z) using reverse communicaton with ZLACON. ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are subpencils of two matrix pairs in ZTGSYL.
[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 COMPLEX*16 array, dimension (LDA, M) On entry, A contains an upper triangular matrix. 
[in]  LDA  LDA is INTEGER The leading dimension of the matrix A. LDA >= max(1, M). 
[in]  B  B is COMPLEX*16 array, dimension (LDB, N) On entry, B contains an upper triangular matrix. 
[in]  LDB  LDB is INTEGER The leading dimension of the matrix B. LDB >= max(1, N). 
[in,out]  C  C is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 ZTGSYL, 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 ZTGSY2 is called by ZTGSYL. 
[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 ZTGSY2 is called by ZTGSYL. 
[out]  INFO  INFO is INTEGER On exit, if INFO is set to =0: Successful exit <0: If INFO = i, input argument number i is illegal. >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues. 
Definition at line 258 of file ztgsy2.f.