Provided by: liblapack-doc_3.7.1-4ubuntu1_all

**NAME**

realSYauxiliary

**SYNOPSIS**

Functionsreal functionslansy(NORM, UPLO, N, A, LDA, WORK)SLANSYreturns 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. subroutineslaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)SLAQSYscales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. subroutineslasy2(LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO)SLASY2solves the Sylvester matrix equation where the matrices are of order 1 or 2. subroutinessyswapr(UPLO, N, A, LDA, I1, I2)SSYSWAPRapplies an elementary permutation on the rows and columns of a symmetric matrix. subroutinestgsy2(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO)STGSY2solves the generalized Sylvester equation (unblocked algorithm).

**Detailed** **Description**

This is the group of real auxiliary functions for SY matrices

**Function** **Documentation**

realfunctionslansy(characterNORM,characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,real,dimension(*)WORK)SLANSYreturns 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:SLANSY 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:SLANSY SLANSY = ( 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 SLANSY 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, SLANSY is set to zero.AA is REAL 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 REAL 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 2016subroutineslaqsy(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,real,dimension(*)S,realSCOND,realAMAX,characterEQUED)SLAQSYscales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.Purpose:SLAQSY 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 REAL 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 REAL array, dimension (N) The scale factors for A.SCONDSCOND is REAL Ratio of the smallest S(i) to the largest S(i).AMAXAMAX is REAL 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 2016subroutineslasy2(logicalLTRANL,logicalLTRANR,integerISGN,integerN1,integerN2,real,dimension(ldtl,*)TL,integerLDTL,real,dimension(ldtr,*)TR,integerLDTR,real,dimension(ldb,*)B,integerLDB,realSCALE,real,dimension(ldx,*)X,integerLDX,realXNORM,integerINFO)SLASY2solves the Sylvester matrix equation where the matrices are of order 1 or 2.Purpose:SLASY2 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:LTRANLLTRANL is LOGICAL On entry, LTRANL specifies the op(TL): = .FALSE., op(TL) = TL, = .TRUE., op(TL) = TL**T.LTRANRLTRANR is LOGICAL On entry, LTRANR specifies the op(TR): = .FALSE., op(TR) = TR, = .TRUE., op(TR) = TR**T.ISGNISGN is INTEGER On entry, ISGN specifies the sign of the equation as described before. ISGN may only be 1 or -1.N1N1 is INTEGER On entry, N1 specifies the order of matrix TL. N1 may only be 0, 1 or 2.N2N2 is INTEGER On entry, N2 specifies the order of matrix TR. N2 may only be 0, 1 or 2.TLTL is REAL array, dimension (LDTL,2) On entry, TL contains an N1 by N1 matrix.LDTLLDTL is INTEGER The leading dimension of the matrix TL. LDTL >= max(1,N1).TRTR is REAL array, dimension (LDTR,2) On entry, TR contains an N2 by N2 matrix.LDTRLDTR is INTEGER The leading dimension of the matrix TR. LDTR >= max(1,N2).BB is REAL array, dimension (LDB,2) On entry, the N1 by N2 matrix B contains the right-hand side of the equation.LDBLDB is INTEGER The leading dimension of the matrix B. LDB >= max(1,N1).SCALESCALE is REAL On exit, SCALE contains the scale factor. SCALE is chosen less than or equal to 1 to prevent the solution overflowing.XX is REAL array, dimension (LDX,2) On exit, X contains the N1 by N2 solution.LDXLDX is INTEGER The leading dimension of the matrix X. LDX >= max(1,N1).XNORMXNORM is REAL On exit, XNORM is the infinity-norm of the solution.INFOINFO 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 Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinessyswapr(characterUPLO,integerN,real,dimension(lda,n)A,integerLDA,integerI1,integerI2)SSYSWAPRapplies an elementary permutation on the rows and columns of a symmetric matrix.Purpose:SSYSWAPR 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 REAL 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 SSYTRF. 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 2016subroutinestgsy2(characterTRANS,integerIJOB,integerM,integerN,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB,real,dimension(ldc,*)C,integerLDC,real,dimension(ldd,*)D,integerLDD,real,dimension(lde,*)E,integerLDE,real,dimension(ldf,*)F,integerLDF,realSCALE,realRDSUM,realRDSCAL,integer,dimension(*)IWORK,integerPQ,integerINFO)STGSY2solves the generalized Sylvester equation (unblocked algorithm).Purpose:STGSY2 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 communicaton with SLACON. STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL 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 STGSYL. See STGSYL for details.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. (SGECON 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 REAL array, dimension (LDA, M) On entry, A contains an upper quasi triangular matrix.LDALDA is INTEGER The leading dimension of the matrix A. LDA >= max(1, M).BB is REAL array, dimension (LDB, N) On entry, B contains an upper quasi triangular matrix.LDBLDB is INTEGER The leading dimension of the matrix B. LDB >= max(1, N).CC is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL On entry, the sum of squares of computed contributions to the Dif-estimate under computation by STGSYL, 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 STGSY2 is called by STGSYL.RDSCALRDSCAL is REAL 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 STGSY2 is called by STGSYL.IWORKIWORK is INTEGER array, dimension (M+N+2)PQPQ is INTEGER On exit, the number of subsystems (of size 2-by-2, 4-by-4 and 8-by-8) solved by this routine.INFOINFO 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 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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