trsyl3 - trsyl3: Sylvester equation, level 3

Generated automatically by Doxygen for LAPACK from the source code. Version 3.12.0 Sun Jul 20 2025 01:40:05 trsyl3(3)

subroutinectrsyl3(charactertrana,charactertranb,integerisgn,integerm,integern,complex,dimension(lda,*)a,integerlda,complex,dimension(ldb,*)b,integerldb,complex,dimension(ldc,*)c,integerldc,realscale,real,dimension(ldswork,*)swork,integerldswork,integerinfo)CTRSYL3Purpose: CTRSYL3 solves the complex Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or op(A)*X - X*op(B) = scale*C, where op(A) = A or A**H, and A and B are both upper triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. This is the block version of the algorithm. ParametersTRANA TRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'C': op(A) = A**H (Conjugate transpose) TRANB TRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'C': op(B) = B**H (Conjugate transpose) ISGN ISGN is INTEGER Specifies the sign in the equation: = +1: solve op(A)*X + X*op(B) = scale*C = -1: solve op(A)*X - X*op(B) = scale*C M M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0. N N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0. A A is COMPLEX array, dimension (LDA,M) The upper triangular matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is COMPLEX array, dimension (LDB,N) The upper triangular matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). C C is COMPLEX array, dimension (LDC,N) On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M) SCALE SCALE is REAL The scale factor, scale, set <= 1 to avoid overflow in X. SWORK SWORK is REAL array, dimension (MAX(2, ROWS), MAX(1,COLS)). On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS and SWORK(2) returns the optimal COLS. LDSWORK LDSWORK is INTEGER LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1) and NB is the optimal block size. If LDSWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the SWORK matrix, returns these values as the first and second entry of the SWORK matrix, and no error message related LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged). subroutinedtrsyl3(charactertrana,charactertranb,integerisgn,integerm,integern,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecision,dimension(ldc,*)c,integerldc,doubleprecisionscale,integer,dimension(*)iwork,integerliwork,doubleprecision,dimension(ldswork,*)swork,integerldswork,integerinfo)DTRSYL3Purpose: DTRSYL3 solves the real Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or op(A)*X - X*op(B) = scale*C, where op(A) = A or A**T, and A and B are both upper quasi- triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. A and B must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. This is the block version of the algorithm. ParametersTRANA TRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'T': op(A) = A**T (Transpose) = 'C': op(A) = A**H (Conjugate transpose = Transpose) TRANB TRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'T': op(B) = B**T (Transpose) = 'C': op(B) = B**H (Conjugate transpose = Transpose) ISGN ISGN is INTEGER Specifies the sign in the equation: = +1: solve op(A)*X + X*op(B) = scale*C = -1: solve op(A)*X - X*op(B) = scale*C M M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0. N N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0. A A is DOUBLE PRECISION array, dimension (LDA,M) The upper quasi-triangular matrix A, in Schur canonical form. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is DOUBLE PRECISION array, dimension (LDB,N) The upper quasi-triangular matrix B, in Schur canonical form. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). C C is DOUBLE PRECISION array, dimension (LDC,N) On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M) SCALE SCALE is DOUBLE PRECISION The scale factor, scale, set <= 1 to avoid overflow in X. IWORK IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK IWORK is INTEGER The dimension of the array IWORK. LIWORK >= ((M + NB - 1) / NB + 1) + ((N + NB - 1) / NB + 1), where NB is the optimal block size. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimension of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. SWORK SWORK is DOUBLE PRECISION array, dimension (MAX(2, ROWS), MAX(1,COLS)). On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS and SWORK(2) returns the optimal COLS. LDSWORK LDSWORK is INTEGER LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1) and NB is the optimal block size. If LDSWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the SWORK matrix, returns these values as the first and second entry of the SWORK matrix, and no error message related LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged). subroutinestrsyl3(charactertrana,charactertranb,integerisgn,integerm,integern,real,dimension(lda,*)a,integerlda,real,dimension(ldb,*)b,integerldb,real,dimension(ldc,*)c,integerldc,realscale,integer,dimension(*)iwork,integerliwork,real,dimension(ldswork,*)swork,integerldswork,integerinfo)STRSYL3Purpose: STRSYL3 solves the real Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or op(A)*X - X*op(B) = scale*C, where op(A) = A or A**T, and A and B are both upper quasi- triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. A and B must be in Schur canonical form (as returned by SHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. This is the block version of the algorithm. ParametersTRANA TRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'T': op(A) = A**T (Transpose) = 'C': op(A) = A**H (Conjugate transpose = Transpose) TRANB TRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'T': op(B) = B**T (Transpose) = 'C': op(B) = B**H (Conjugate transpose = Transpose) ISGN ISGN is INTEGER Specifies the sign in the equation: = +1: solve op(A)*X + X*op(B) = scale*C = -1: solve op(A)*X - X*op(B) = scale*C M M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0. N N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0. A A is REAL array, dimension (LDA,M) The upper quasi-triangular matrix A, in Schur canonical form. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is REAL array, dimension (LDB,N) The upper quasi-triangular matrix B, in Schur canonical form. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). C C is REAL array, dimension (LDC,N) On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M) SCALE SCALE is REAL The scale factor, scale, set <= 1 to avoid overflow in X. IWORK IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK IWORK is INTEGER The dimension of the array IWORK. LIWORK >= ((M + NB - 1) / NB + 1) + ((N + NB - 1) / NB + 1), where NB is the optimal block size. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimension of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. SWORK SWORK is REAL array, dimension (MAX(2, ROWS), MAX(1,COLS)). On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS and SWORK(2) returns the optimal COLS. LDSWORK LDSWORK is INTEGER LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1) and NB is the optimal block size. If LDSWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the SWORK matrix, returns these values as the first and second entry of the SWORK matrix, and no error message related LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged). subroutineztrsyl3(charactertrana,charactertranb,integerisgn,integerm,integern,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldb,*)b,integerldb,complex*16,dimension(ldc,*)c,integerldc,doubleprecisionscale,doubleprecision,dimension(ldswork,*)swork,integerldswork,integerinfo)ZTRSYL3Purpose: ZTRSYL3 solves the complex Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or op(A)*X - X*op(B) = scale*C, where op(A) = A or A**H, and A and B are both upper triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. This is the block version of the algorithm. ParametersTRANA TRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'C': op(A) = A**H (Conjugate transpose) TRANB TRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'C': op(B) = B**H (Conjugate transpose) ISGN ISGN is INTEGER Specifies the sign in the equation: = +1: solve op(A)*X + X*op(B) = scale*C = -1: solve op(A)*X - X*op(B) = scale*C M M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0. N N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0. A A is COMPLEX*16 array, dimension (LDA,M) The upper triangular matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is COMPLEX*16 array, dimension (LDB,N) The upper triangular matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). C C is COMPLEX*16 array, dimension (LDC,N) On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M) SCALE SCALE is DOUBLE PRECISION The scale factor, scale, set <= 1 to avoid overflow in X. SWORK SWORK is DOUBLE PRECISION array, dimension (MAX(2, ROWS), MAX(1,COLS)). On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS and SWORK(2) returns the optimal COLS. LDSWORK LDSWORK is INTEGER LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1) and NB is the optimal block size. If LDSWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the SWORK matrix, returns these values as the first and second entry of the SWORK matrix, and no error message related LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged).

trsyl3 - trsyl3: Sylvester equation, level 3

Functions subroutine ctrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, swork, ldswork, info) CTRSYL3 subroutine dtrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, iwork, liwork, swork, ldswork, info) DTRSYL3 subroutine strsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, iwork, liwork, swork, ldswork, info) STRSYL3 subroutine ztrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, swork, ldswork, info) ZTRSYL3