.TH "complexGTcomputational" 3 "Sun Nov 27 2022" "Version 3.11.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME complexGTcomputational \- complex .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBcgtcon\fP (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)" .br .RI "\fBCGTCON\fP " .ti -1c .RI "subroutine \fBcgtrfs\fP (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)" .br .RI "\fBCGTRFS\fP " .ti -1c .RI "subroutine \fBcgttrf\fP (N, DL, D, DU, DU2, IPIV, INFO)" .br .RI "\fBCGTTRF\fP " .ti -1c .RI "subroutine \fBcgttrs\fP (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)" .br .RI "\fBCGTTRS\fP " .ti -1c .RI "subroutine \fBcgtts2\fP (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)" .br .RI "\fBCGTTS2\fP solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf\&. " .in -1c .SH "Detailed Description" .PP This is the group of complex computational functions for GT matrices .SH "Function Documentation" .PP .SS "subroutine cgtcon (character NORM, integer N, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( * ) DU2, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)" .PP \fBCGTCON\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CGTCON estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by CGTTRF\&. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A)))\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fINORM\fP .PP .nf NORM is CHARACTER*1 Specifies whether the 1-norm condition number or the infinity-norm condition number is required: = '1' or 'O': 1-norm; = 'I': Infinity-norm\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by CGTTRF\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second superdiagonal of U\&. .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 was interchanged with row IPIV(i)\&. IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required\&. .fi .PP .br \fIANORM\fP .PP .nf ANORM is REAL If NORM = '1' or 'O', the 1-norm of the original matrix A\&. If NORM = 'I', the infinity-norm of the original matrix A\&. .fi .PP .br \fIRCOND\fP .PP .nf RCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX array, dimension (2*N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .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 .SS "subroutine cgtrfs (character TRANS, integer N, integer NRHS, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( * ) DLF, complex, dimension( * ) DF, complex, dimension( * ) DUF, complex, dimension( * ) DU2, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)" .PP \fBCGTRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX array, dimension (N-1) The (n-1) subdiagonal elements of A\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension (N) The diagonal elements of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX array, dimension (N-1) The (n-1) superdiagonal elements of A\&. .fi .PP .br \fIDLF\fP .PP .nf DLF is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by CGTTRF\&. .fi .PP .br \fIDF\fP .PP .nf DF is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDUF\fP .PP .nf DUF is COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second superdiagonal of U\&. .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 was interchanged with row IPIV(i)\&. IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX array, dimension (LDB,NRHS) The right hand side matrix B\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIX\fP .PP .nf X is COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CGTTRS\&. On exit, the improved solution matrix X\&. .fi .PP .br \fILDX\fP .PP .nf LDX is INTEGER The leading dimension of the array X\&. LDX >= max(1,N)\&. .fi .PP .br \fIFERR\fP .PP .nf FERR is REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X)\&. If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j)\&. The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error\&. .fi .PP .br \fIBERR\fP .PP .nf BERR is REAL array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i\&.e\&., the smallest relative change in any element of A or B that makes X(j) an exact solution)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX array, dimension (2*N) .fi .PP .br \fIRWORK\fP .PP .nf RWORK is REAL array, dimension (N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf ITMAX is the maximum number of steps of iterative refinement\&. .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 .SS "subroutine cgttrf (integer N, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( * ) DU2, integer, dimension( * ) IPIV, integer INFO)" .PP \fBCGTTRF\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CGTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges\&. The factorization has the form A = L * U where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) sub-diagonal elements of A\&. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension (N) On entry, D must contain the diagonal elements of A\&. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) super-diagonal elements of A\&. On exit, DU is overwritten by the (n-1) elements of the first super-diagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX array, dimension (N-2) On exit, DU2 is overwritten by the (n-2) elements of the second super-diagonal of U\&. .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 was interchanged with row IPIV(i)\&. IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, U(k,k) is exactly zero\&. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations\&. .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 .SS "subroutine cgttrs (character TRANS, integer N, integer NRHS, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( * ) DU2, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)" .PP \fBCGTTRS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CGTTRS solves one of the systems of equations A * X = B, A**T * X = B, or A**H * X = B, with a tridiagonal matrix A using the LU factorization computed by CGTTRF\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf TRANS is CHARACTER*1 Specifies the form of the system of equations\&. = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX array, dimension (N-1) The (n-1) elements of the first super-diagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second super-diagonal of U\&. .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 was interchanged with row IPIV(i)\&. IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX array, dimension (LDB,NRHS) On entry, the matrix of right hand side vectors B\&. On exit, B is overwritten by the solution vectors X\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value .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 .SS "subroutine cgtts2 (integer ITRANS, integer N, integer NRHS, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( * ) DU2, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB)" .PP \fBCGTTS2\fP solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CGTTS2 solves one of the systems of equations A * X = B, A**T * X = B, or A**H * X = B, with a tridiagonal matrix A using the LU factorization computed by CGTTRF\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIITRANS\fP .PP .nf ITRANS is INTEGER Specifies the form of the system of equations\&. = 0: A * X = B (No transpose) = 1: A**T * X = B (Transpose) = 2: A**H * X = B (Conjugate transpose) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX array, dimension (N-1) The (n-1) elements of the first super-diagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second super-diagonal of U\&. .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 was interchanged with row IPIV(i)\&. IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX array, dimension (LDB,NRHS) On entry, the matrix of right hand side vectors B\&. On exit, B is overwritten by the solution vectors X\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .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 .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.