trsyl3(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

✨ NAME ✨

trsyl3 — okay so this is trsyl3: the Sylvester equation, but make it level 3 💅 (that's the trsyl3(3) manual page, bestie, in case the header vibe check wasn't clear 🎀)

🦋 SYNOPSIS 🦋

💖 Functions 💖

Four routines, all siblings, all solving the Sylvester equation like the icons they are 👑

  • ctrsyl3 — the complex one 🌸
subroutine ctrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc,
    scale, swork, ldswork, info)

CTRSYL3

  • dtrsyl3 — the double precision one ✨
subroutine dtrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc,
    scale, iwork, liwork, swork, ldswork, info)

DTRSYL3

  • strsyl3 — the single precision one 🍓
subroutine strsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc,
    scale, iwork, liwork, swork, ldswork, info)

STRSYL3

  • ztrsyl3 — the double complex one 🦋
subroutine ztrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc,
    scale, swork, ldswork, info)

ZTRSYL3

🌈 Detailed Description 🌈

(yes that's the section name, don't come for me)

👑 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 — main character energy, complex edition 💜

✨ Purpose ✨

CTRSYL3 solves the complex Sylvester matrix equation, and it's literally giving:

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. Periodt. 🔥

🎀 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 — so don't come crying when your RHS is gone, that's the whole point 💀
  • 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).
⚠️ Heads up: that INFO = 1 case is not a drill — the routine fudged the values to get you an answer. Still INFO = 1 though, so read it, don't skip it.

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 — double precision, double the drama 💅

✨ 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. No cap, that's just the requirement 😭

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 — yes bestie, it eats your input, that's the deal 🙈
  • 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 — LIWORK 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 — single precision, same energy 🌸

✨ Purpose ✨

(and this is exactly where part 1 hands the mic off — STRSYL3's Purpose block picks up the second this page continues, so stay tuned bestie, we are not done here 💖)

So 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 — 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 — 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 — 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 — INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
  • N — INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
  • A — REAL array, dimension (LDA,M). The upper quasi-triangular matrix A, in Schur canonical form.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — REAL array, dimension (LDB,N). The upper quasi-triangular matrix B, in Schur canonical form.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • C — 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 — INTEGER. The leading dimension of the array C. LDC >= max(1,M)
  • SCALE — REAL. The scale factor, scale, set <= 1 to avoid overflow in X.
  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK — IWORK is INTEGER (yes, the original really says that 😭). 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 — 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 — 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 — 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 🦋

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 — CHARACTER*1. Specifies the option op(A):
    • = 'N': op(A) = A (No transpose)
    • = 'C': op(A) = A**H (Conjugate transpose)
  • TRANB — CHARACTER*1. Specifies the option op(B):
    • = 'N': op(B) = B (No transpose)
    • = 'C': op(B) = B**H (Conjugate transpose)
  • ISGN — 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 — INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
  • N — INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
  • A — COMPLEX*16 array, dimension (LDA,M). The upper triangular matrix A.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — COMPLEX*16 array, dimension (LDB,N). The upper triangular matrix B.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • C — 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 — INTEGER. The leading dimension of the array C. LDC >= max(1,M)
  • SCALE — DOUBLE PRECISION. The scale factor, scale, set <= 1 to avoid overflow in X.
  • SWORK — 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 — 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 — 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 🌈

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💖 That's the whole page, bestie! This little icon is LAPACK Version 3.12.0, and you're reading trsyl3(3). Go slay those Sylvester equations — no cap! 🎀👑
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