---
name: "trsyl3"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/trsyl3"
---

# trsyl3(3)

trsyl3: Sylvester equation, level 3

## NAME

trsyl3 - trsyl3: Sylvester equation, level 3

## SYNOPSIS

### 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**

## Detailed Description

## Function Documentation

### subroutine ctrsyl3 (character trana, character tranb, integer isgn, integer m, integer n, complex, dimension( lda, \* ) a, integer lda, complex, dimension( ldb, \* ) b, integer ldb, complex, dimension( ldc, \* ) c, integer ldc, real scale, real, dimension( ldswork, \* ) swork, integer ldswork, integer info)

**CTRSYL3**

**Purpose:**

```
!>
!>  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.
!>
```

**Parameters**

*TRANA*

```
!>          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).
!>
```

Definition at line **167** of file **ctrsyl3.f**.

### subroutine dtrsyl3 (character trana, character tranb, integer isgn, 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 scale, integer, dimension( \* ) iwork, integer liwork, double precision, dimension( ldswork, \* ) swork, integer ldswork, integer info)

**DTRSYL3**

**Purpose:**

```
!>
!>  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.
!>
```

**Parameters**

*TRANA*

```
!>          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).
!>
```

Definition at line **197** of file **dtrsyl3.f**.

### subroutine strsyl3 (character trana, character tranb, integer isgn, integer m, integer n, real, dimension( lda, \* ) a, integer lda, real, dimension( ldb, \* ) b, integer ldb, real, dimension( ldc, \* ) c, integer ldc, real scale, integer, dimension( \* ) iwork, integer liwork, real, dimension( ldswork, \* ) swork, integer ldswork, integer info)

**STRSYL3**

**Purpose:**

```
!>
!>  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.
!>
```

**Parameters**

*TRANA*

```
!>          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).
!>
```

Definition at line **196** of file **strsyl3.f**.

### subroutine ztrsyl3 (character trana, character tranb, integer isgn, 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, double precision scale, double precision, dimension( ldswork, \* ) swork, integer ldswork, integer info)

**ZTRSYL3**

**Purpose:**

```
!>
!>  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.
!>
```

**Parameters**

*TRANA*

```
!>          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).
!>
```

Definition at line **168** of file **ztrsyl3.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.