Provided by: liblapack-doc_3.7.1-4ubuntu1_all

**NAME**

complex16SYauxiliary

**SYNOPSIS**

Functionssubroutinezlaesy(A, B, C, RT1, RT2, EVSCAL, CS1, SN1)ZLAESYcomputes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix. double precision functionzlansy(NORM, UPLO, N, A, LDA, WORK)ZLANSYreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix. subroutinezlaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)ZLAQSYscales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. subroutinezsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)ZSYMVcomputes a matrix-vector product for a complex symmetric matrix. subroutinezsyr(UPLO, N, ALPHA, X, INCX, A, LDA)ZSYRperforms the symmetric rank-1 update of a complex symmetric matrix. subroutinezsyswapr(UPLO, N, A, LDA, I1, I2)ZSYSWAPRsubroutineztgsy2(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)ZTGSY2solves the generalized Sylvester equation (unblocked algorithm).

**Detailed** **Description**

This is the group of complex16 auxiliary functions for SY matrices

**Function** **Documentation**

subroutinezlaesy(complex*16A,complex*16B,complex*16C,complex*16RT1,complex*16RT2,complex*16EVSCAL,complex*16CS1,complex*16SN1)ZLAESYcomputes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.Purpose:ZLAESY computes the eigendecomposition of a 2-by-2 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 ]Parameters:AA is COMPLEX*16 The ( 1, 1 ) element of input matrix.BB is COMPLEX*16 The ( 1, 2 ) element of input matrix. The ( 2, 1 ) element is also given by B, since the 2-by-2 matrix is symmetric.CC is COMPLEX*16 The ( 2, 2 ) element of input matrix.RT1RT1 is COMPLEX*16 The eigenvalue of larger modulus.RT2RT2 is COMPLEX*16 The eigenvalue of smaller modulus.EVSCALEVSCAL 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 2-by-2 matrix could not be diagonalized, or the norm of the matrix of eigenvectors before scaling was larger than the threshold value THRESH (set below).CS1CS1 is COMPLEX*16SN1SN1 is COMPLEX*16 If EVSCAL .NE. 0, ( CS1, SN1 ) is the unit right eigenvector for RT1.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctionzlansy(characterNORM,characterUPLO,integerN,complex*16,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)WORK)ZLANSYreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.Purpose: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.Returns:ZLANSY 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in ZLANSY as described above.UPLOUPLO 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 referencedNN is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANSY is set to zero.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(N,1).WORKWORK 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 Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinezlaqsy(characterUPLO,integerN,complex*16,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)S,doubleprecisionSCOND,doubleprecisionAMAX,characterEQUED)ZLAQSYscales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.Purpose:ZLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrix A. N >= 0.AA 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).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(N,1).SS is DOUBLE PRECISION array, dimension (N) The scale factors for A.SCONDSCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).AMAXAMAX is DOUBLE PRECISION Absolute value of largest matrix entry.EQUEDEQUED 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).InternalParameters: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 Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinezsymv(characterUPLO,integerN,complex*16ALPHA,complex*16,dimension(lda,*)A,integerLDA,complex*16,dimension(*)X,integerINCX,complex*16BETA,complex*16,dimension(*)Y,integerINCY)ZSYMVcomputes a matrix-vector product for a complex symmetric matrix.Purpose:ZSYMV performs the matrix-vector 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.Parameters:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit.ALPHAALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit.AA 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.LDALDA 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.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.BETABETA 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.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinezsyr(characterUPLO,integerN,complex*16ALPHA,complex*16,dimension(*)X,integerINCX,complex*16,dimension(lda,*)A,integerLDA)ZSYRperforms the symmetric rank-1 update of a complex symmetric matrix.Purpose: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.Parameters:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit.ALPHAALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.AA 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.LDALDA 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.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinezsyswapr(characterUPLO,integerN,complex*16,dimension(lda,n)A,integerLDA,integerI1,integerI2)ZSYSWAPRPurpose:ZSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix.Parameters:UPLOUPLO 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.NN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).I1I1 is INTEGER Index of the first row to swapI2I2 is INTEGER Index of the second row to swapAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutineztgsy2(characterTRANS,integerIJOB,integerM,integerN,complex*16,dimension(lda,*)A,integerLDA,complex*16,dimension(ldb,*)B,integerLDB,complex*16,dimension(ldc,*)C,integerLDC,complex*16,dimension(ldd,*)D,integerLDD,complex*16,dimension(lde,*)E,integerLDE,complex*16,dimension(ldf,*)F,integerLDF,doubleprecisionSCALE,doubleprecisionRDSUM,doubleprecisionRDSCAL,integerINFO)ZTGSY2solves the generalized Sylvester equation (unblocked algorithm).Purpose: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 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. 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 sub-pencils of two matrix pairs in ZTGSYL.Parameters:TRANSTRANS is CHARACTER*1 = 'N', solve the generalized Sylvester equation (1). = 'T': solve the 'transposed' system (3).IJOBIJOB 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'.MM is INTEGER On entry, M specifies the order of A and D, and the row dimension of C, F, R and L.NN is INTEGER On entry, N specifies the order of B and E, and the column dimension of C, F, R and L.AA is COMPLEX*16 array, dimension (LDA, M) On entry, A contains an upper triangular matrix.LDALDA is INTEGER The leading dimension of the matrix A. LDA >= max(1, M).BB is COMPLEX*16 array, dimension (LDB, N) On entry, B contains an upper triangular matrix.LDBLDB is INTEGER The leading dimension of the matrix B. LDB >= max(1, N).CC is COMPLEX*16 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.LDCLDC is INTEGER The leading dimension of the matrix C. LDC >= max(1, M).DD is COMPLEX*16 array, dimension (LDD, M) On entry, D contains an upper triangular matrix.LDDLDD is INTEGER The leading dimension of the matrix D. LDD >= max(1, M).EE is COMPLEX*16 array, dimension (LDE, N) On entry, E contains an upper triangular matrix.LDELDE is INTEGER The leading dimension of the matrix E. LDE >= max(1, N).FF is COMPLEX*16 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.LDFLDF is INTEGER The leading dimension of the matrix F. LDF >= max(1, M).SCALESCALE 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.RDSUMRDSUM is DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Dif-estimate 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 sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when ZTGSY2 is called by ZTGSYL.RDSCALRDSCAL 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.INFOINFO 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.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

**Author**

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