--- name: "latdf" 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/latdf" --- # latdf(3) latdf: Dif-estimate with complete pivoting LU, step in tgsen ## NAME latdf - latdf: Dif-estimate with complete pivoting LU, step in tgsen ## SYNOPSIS ### Functions subroutine **clatdf** (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)\ **CLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine **dlatdf** (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)\ **DLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine **slatdf** (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)\ **SLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine **zlatdf** (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)\ **ZLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. ## Detailed Description ## Function Documentation ### subroutine clatdf (integer ijob, integer n, complex, dimension( ldz, \* ) z, integer ldz, complex, dimension( \* ) rhs, real rdsum, real rdscal, integer, dimension( \* ) ipiv, integer, dimension( \* ) jpiv) **CLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. **Purpose:** ``` !> !> CLATDF computes the contribution to the reciprocal Dif-estimate !> by solving for x in Z * x = b, where b is chosen such that the norm !> of x is as large as possible. It is assumed that LU decomposition !> of Z has been computed by CGETC2. On entry RHS = f holds the !> contribution from earlier solved sub-systems, and on return RHS = x. !> !> The factorization of Z returned by CGETC2 has the form !> Z = P * L * U * Q, where P and Q are permutation matrices. L is lower !> triangular with unit diagonal elements and U is upper triangular. !> ``` **Parameters** *IJOB* ``` !> IJOB is INTEGER !> IJOB = 2: First compute an approximative null-vector e !> of Z using CGECON, e is normalized and solve for !> Zx = +-e - f with the sign giving the greater value of !> 2-norm(x). About 5 times as expensive as Default. !> IJOB .ne. 2: Local look ahead strategy where !> all entries of the r.h.s. b is chosen as either +1 or !> -1. Default. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrix Z. !> ``` *Z* ``` !> Z is COMPLEX array, dimension (LDZ, N) !> On entry, the LU part of the factorization of the n-by-n !> matrix Z computed by CGETC2: Z = P * L * U * Q !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDA >= max(1, N). !> ``` *RHS* ``` !> RHS is COMPLEX array, dimension (N). !> On entry, RHS contains contributions from other subsystems. !> On exit, RHS contains the solution of the subsystem with !> entries according to the value of IJOB (see above). !> ``` *RDSUM* ``` !> RDSUM is REAL !> On entry, the sum of squares of computed contributions to !> the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by CTGSYL. !> ``` *RDSCAL* ``` !> RDSCAL 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 CTGSY2 is called by !> CTGSYL. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= i <= N, row i of the !> matrix has been interchanged with row IPIV(i). !> ``` *JPIV* ``` !> JPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= j <= N, column j of the !> matrix has been interchanged with column JPIV(j). !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** This routine is a further developed implementation of algorithm BSOLVE in \[1\] using complete pivoting in the LU factorization. **Contributors:** Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. **References:** \[1\] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. \[2\] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. Definition at line **167** of file **clatdf.f**. ### subroutine dlatdf (integer ijob, integer n, double precision, dimension( ldz, \* ) z, integer ldz, double precision, dimension( \* ) rhs, double precision rdsum, double precision rdscal, integer, dimension( \* ) ipiv, integer, dimension( \* ) jpiv) **DLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. **Purpose:** ``` !> !> DLATDF uses the LU factorization of the n-by-n matrix Z computed by !> DGETC2 and computes a contribution to the reciprocal Dif-estimate !> by solving Z * x = b for x, and choosing the r.h.s. b such that !> the norm of x is as large as possible. On entry RHS = b holds the !> contribution from earlier solved sub-systems, and on return RHS = x. !> !> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, !> where P and Q are permutation matrices. L is lower triangular with !> unit diagonal elements and U is upper triangular. !> ``` **Parameters** *IJOB* ``` !> IJOB is INTEGER !> IJOB = 2: First compute an approximative null-vector e !> of Z using DGECON, e is normalized and solve for !> Zx = +-e - f with the sign giving the greater value !> of 2-norm(x). About 5 times as expensive as Default. !> IJOB .ne. 2: Local look ahead strategy where all entries of !> the r.h.s. b is chosen as either +1 or -1 (Default). !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrix Z. !> ``` *Z* ``` !> Z is DOUBLE PRECISION array, dimension (LDZ, N) !> On entry, the LU part of the factorization of the n-by-n !> matrix Z computed by DGETC2: Z = P * L * U * Q !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDA >= max(1, N). !> ``` *RHS* ``` !> RHS is DOUBLE PRECISION array, dimension (N) !> On entry, RHS contains contributions from other subsystems. !> On exit, RHS contains the solution of the subsystem with !> entries according to the value of IJOB (see above). !> ``` *RDSUM* ``` !> RDSUM is DOUBLE PRECISION !> On entry, the sum of squares of computed contributions to !> the Dif-estimate under computation by DTGSYL, 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 DTGSY2 is called by STGSYL. !> ``` *RDSCAL* ``` !> RDSCAL 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 DTGSY2 is called by !> DTGSYL. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= i <= N, row i of the !> matrix has been interchanged with row IPIV(i). !> ``` *JPIV* ``` !> JPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= j <= N, column j of the !> matrix has been interchanged with column JPIV(j). !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** This routine is a further developed implementation of algorithm BSOLVE in \[1\] using complete pivoting in the LU factorization. **Contributors:** Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. **References:** ``` !> !> !> [1] Bo Kagstrom and Lars Westin, !> Generalized Schur Methods with Condition Estimators for !> Solving the Generalized Sylvester Equation, IEEE Transactions !> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. !> !> [2] Peter Poromaa, !> On Efficient and Robust Estimators for the Separation !> between two Regular Matrix Pairs with Applications in !> Condition Estimation. Report IMINF-95.05, Departement of !> Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. !> ``` Definition at line **169** of file **dlatdf.f**. ### subroutine slatdf (integer ijob, integer n, real, dimension( ldz, \* ) z, integer ldz, real, dimension( \* ) rhs, real rdsum, real rdscal, integer, dimension( \* ) ipiv, integer, dimension( \* ) jpiv) **SLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. **Purpose:** ``` !> !> SLATDF uses the LU factorization of the n-by-n matrix Z computed by !> SGETC2 and computes a contribution to the reciprocal Dif-estimate !> by solving Z * x = b for x, and choosing the r.h.s. b such that !> the norm of x is as large as possible. On entry RHS = b holds the !> contribution from earlier solved sub-systems, and on return RHS = x. !> !> The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q, !> where P and Q are permutation matrices. L is lower triangular with !> unit diagonal elements and U is upper triangular. !> ``` **Parameters** *IJOB* ``` !> IJOB is INTEGER !> IJOB = 2: First compute an approximative null-vector e !> of Z using SGECON, e is normalized and solve for !> Zx = +-e - f with the sign giving the greater value !> of 2-norm(x). About 5 times as expensive as Default. !> IJOB .ne. 2: Local look ahead strategy where all entries of !> the r.h.s. b is chosen as either +1 or -1 (Default). !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrix Z. !> ``` *Z* ``` !> Z is REAL array, dimension (LDZ, N) !> On entry, the LU part of the factorization of the n-by-n !> matrix Z computed by SGETC2: Z = P * L * U * Q !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDA >= max(1, N). !> ``` *RHS* ``` !> RHS is REAL array, dimension N. !> On entry, RHS contains contributions from other subsystems. !> On exit, RHS contains the solution of the subsystem with !> entries according to the value of IJOB (see above). !> ``` *RDSUM* ``` !> RDSUM 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. !> ``` *RDSCAL* ``` !> RDSCAL 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. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= i <= N, row i of the !> matrix has been interchanged with row IPIV(i). !> ``` *JPIV* ``` !> JPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= j <= N, column j of the !> matrix has been interchanged with column JPIV(j). !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** This routine is a further developed implementation of algorithm BSOLVE in \[1\] using complete pivoting in the LU factorization. **Contributors:** Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. **References:** ``` !> !> !> [1] Bo Kagstrom and Lars Westin, !> Generalized Schur Methods with Condition Estimators for !> Solving the Generalized Sylvester Equation, IEEE Transactions !> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. !> !> [2] Peter Poromaa, !> On Efficient and Robust Estimators for the Separation !> between two Regular Matrix Pairs with Applications in !> Condition Estimation. Report IMINF-95.05, Departement of !> Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. !> ``` Definition at line **169** of file **slatdf.f**. ### subroutine zlatdf (integer ijob, integer n, complex\*16, dimension( ldz, \* ) z, integer ldz, complex\*16, dimension( \* ) rhs, double precision rdsum, double precision rdscal, integer, dimension( \* ) ipiv, integer, dimension( \* ) jpiv) **ZLATDF** uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. **Purpose:** ``` !> !> ZLATDF computes the contribution to the reciprocal Dif-estimate !> by solving for x in Z * x = b, where b is chosen such that the norm !> of x is as large as possible. It is assumed that LU decomposition !> of Z has been computed by ZGETC2. On entry RHS = f holds the !> contribution from earlier solved sub-systems, and on return RHS = x. !> !> The factorization of Z returned by ZGETC2 has the form !> Z = P * L * U * Q, where P and Q are permutation matrices. L is lower !> triangular with unit diagonal elements and U is upper triangular. !> ``` **Parameters** *IJOB* ``` !> IJOB is INTEGER !> IJOB = 2: First compute an approximative null-vector e !> of Z using ZGECON, e is normalized and solve for !> Zx = +-e - f with the sign giving the greater value of !> 2-norm(x). About 5 times as expensive as Default. !> IJOB .ne. 2: Local look ahead strategy where !> all entries of the r.h.s. b is chosen as either +1 or !> -1. Default. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrix Z. !> ``` *Z* ``` !> Z is COMPLEX*16 array, dimension (LDZ, N) !> On entry, the LU part of the factorization of the n-by-n !> matrix Z computed by ZGETC2: Z = P * L * U * Q !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDA >= max(1, N). !> ``` *RHS* ``` !> RHS is COMPLEX*16 array, dimension (N). !> On entry, RHS contains contributions from other subsystems. !> On exit, RHS contains the solution of the subsystem with !> entries according to the value of IJOB (see above). !> ``` *RDSUM* ``` !> RDSUM 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 CTGSYL. !> ``` *RDSCAL* ``` !> RDSCAL 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. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= i <= N, row i of the !> matrix has been interchanged with row IPIV(i). !> ``` *JPIV* ``` !> JPIV is INTEGER array, dimension (N). !> The pivot indices; for 1 <= j <= N, column j of the !> matrix has been interchanged with column JPIV(j). !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** This routine is a further developed implementation of algorithm BSOLVE in \[1\] using complete pivoting in the LU factorization. **Contributors:** Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. **References:** \[1\] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. \ \ \[2\] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. Definition at line **167** of file **zlatdf.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.