SGTCON estimates the reciprocal of the condition number of a real


 tridiagonal matrix A using the LU factorization as computed by


 SGTTRF.


 An estimate is obtained for norm(inv(A)), and the reciprocal of the


 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

NORM

          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.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

DL

          DL is REAL array, dimension (N-1)


          The (n-1) multipliers that define the matrix L from the


          LU factorization of A as computed by SGTTRF.

D

          D is REAL array, dimension (N)


          The n diagonal elements of the upper triangular matrix U from


          the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)


          The (n-1) elements of the first superdiagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)


          The (n-2) elements of the second superdiagonal of U.

IPIV

          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.

ANORM

          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.

RCOND

          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.

WORK

          WORK is REAL array, dimension (2*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

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NAG Ltd.

 SGTRFS 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.

TRANS

          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 = Transpose)

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

DL

          DL is REAL array, dimension (N-1)


          The (n-1) subdiagonal elements of A.

D

          D is REAL array, dimension (N)


          The diagonal elements of A.

DU

          DU is REAL array, dimension (N-1)


          The (n-1) superdiagonal elements of A.

DLF

          DLF is REAL array, dimension (N-1)


          The (n-1) multipliers that define the matrix L from the


          LU factorization of A as computed by SGTTRF.

DF

          DF is REAL array, dimension (N)


          The n diagonal elements of the upper triangular matrix U from


          the LU factorization of A.

DUF

          DUF is REAL array, dimension (N-1)


          The (n-1) elements of the first superdiagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)


          The (n-2) elements of the second superdiagonal of U.

IPIV

          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.

B

          B is REAL array, dimension (LDB,NRHS)


          The right hand side matrix B.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is REAL array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by SGTTRS.


          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          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.

BERR

          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).

WORK

          WORK is REAL array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

  ITMAX is the maximum number of steps of iterative refinement.

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NAG Ltd.

 SGTTRF computes an LU factorization of a real 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.

N

          N is INTEGER


          The order of the matrix A.

DL

          DL is REAL 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.

D

          D is REAL 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.

DU

          DU is REAL 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.

DU2

          DU2 is REAL array, dimension (N-2)


          On exit, DU2 is overwritten by the (n-2) elements of the


          second super-diagonal of U.

IPIV

          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.

INFO

          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.

Univ. of Tennessee

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Univ. of Colorado Denver

NAG Ltd.

 SGTTRS solves one of the systems of equations


    A*X = B  or  A**T*X = B,


 with a tridiagonal matrix A using the LU factorization computed


 by SGTTRF.

TRANS

          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**T* X = B  (Conjugate transpose = Transpose)

N

          N is INTEGER


          The order of the matrix A.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

DL

          DL is REAL array, dimension (N-1)


          The (n-1) multipliers that define the matrix L from the


          LU factorization of A.

D

          D is REAL array, dimension (N)


          The n diagonal elements of the upper triangular matrix U from


          the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)


          The (n-1) elements of the first super-diagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)


          The (n-2) elements of the second super-diagonal of U.

IPIV

          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.

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the matrix of right hand side vectors B.


          On exit, B is overwritten by the solution vectors X.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

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NAG Ltd.

 SGTTS2 solves one of the systems of equations


    A*X = B  or  A**T*X = B,


 with a tridiagonal matrix A using the LU factorization computed


 by SGTTRF.

ITRANS

          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**T* X = B  (Conjugate transpose = Transpose)

N

          N is INTEGER


          The order of the matrix A.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

DL

          DL is REAL array, dimension (N-1)


          The (n-1) multipliers that define the matrix L from the


          LU factorization of A.

D

          D is REAL array, dimension (N)


          The n diagonal elements of the upper triangular matrix U from


          the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)


          The (n-1) elements of the first super-diagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)


          The (n-2) elements of the second super-diagonal of U.

IPIV

          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.

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the matrix of right hand side vectors B.


          On exit, B is overwritten by the solution vectors X.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.