.TH "latdf" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME latdf \- latdf: Dif-estimate with complete pivoting LU, step in tgsen .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclatdf\fP (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)" .br .RI "\fBCLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. " .ti -1c .RI "subroutine \fBdlatdf\fP (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)" .br .RI "\fBDLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. " .ti -1c .RI "subroutine \fBslatdf\fP (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)" .br .RI "\fBSLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. " .ti -1c .RI "subroutine \fBzlatdf\fP (ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv)" .br .RI "\fBZLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "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)" .PP \fBCLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIIJOB\fP .PP .nf 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\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix Z\&. .fi .PP .br \fIZ\fP .PP .nf 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 .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. LDA >= max(1, N)\&. .fi .PP .br \fIRHS\fP .PP .nf 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)\&. .fi .PP .br \fIRDSUM\fP .PP .nf 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\&. .fi .PP .br \fIRDSCAL\fP .PP .nf 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\&. .fi .PP .br \fIIPIV\fP .PP .nf 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)\&. .fi .PP .br \fIJPIV\fP .PP .nf 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)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization\&. .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 [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\&. .RE .PP [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\&. .SS "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)" .PP \fBDLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIIJOB\fP .PP .nf 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)\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix Z\&. .fi .PP .br \fIZ\fP .PP .nf 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 .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. LDA >= max(1, N)\&. .fi .PP .br \fIRHS\fP .PP .nf 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)\&. .fi .PP .br \fIRDSUM\fP .PP .nf 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\&. .fi .PP .br \fIRDSCAL\fP .PP .nf 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\&. .fi .PP .br \fIIPIV\fP .PP .nf 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)\&. .fi .PP .br \fIJPIV\fP .PP .nf 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)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization\&. .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 .PP .nf [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\&. .fi .PP .RE .PP .SS "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)" .PP \fBSLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIIJOB\fP .PP .nf 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)\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix Z\&. .fi .PP .br \fIZ\fP .PP .nf 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 .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. LDA >= max(1, N)\&. .fi .PP .br \fIRHS\fP .PP .nf 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)\&. .fi .PP .br \fIRDSUM\fP .PP .nf 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\&. .fi .PP .br \fIRDSCAL\fP .PP .nf 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\&. .fi .PP .br \fIIPIV\fP .PP .nf 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)\&. .fi .PP .br \fIJPIV\fP .PP .nf 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)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization\&. .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 .PP .nf [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\&. .fi .PP .RE .PP .SS "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)" .PP \fBZLATDF\fP uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIIJOB\fP .PP .nf 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\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix Z\&. .fi .PP .br \fIZ\fP .PP .nf 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 .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. LDA >= max(1, N)\&. .fi .PP .br \fIRHS\fP .PP .nf 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)\&. .fi .PP .br \fIRDSUM\fP .PP .nf 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\&. .fi .PP .br \fIRDSCAL\fP .PP .nf 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\&. .fi .PP .br \fIIPIV\fP .PP .nf 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)\&. .fi .PP .br \fIJPIV\fP .PP .nf 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)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization\&. .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 [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\&. .br [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\&. .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.