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dgtsvx(3P)		    Sun Performance Library		    dgtsvx(3P)

NAME
       dgtsvx  -  use  the  LU factorization to compute the solution to a real
       system of linear equations A * X = B or A**T * X = B,

SYNOPSIS
       SUBROUTINE DGTSVX(FACT, TRANSA, N, NRHS, LOW, D, UP, LOWF, DF,
	     UPF1, UPF2, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
	     WORK2, INFO)

       CHARACTER * 1 FACT, TRANSA
       INTEGER N, NRHS, LDB, LDX, INFO
       INTEGER IPIVOT(*), WORK2(*)
       DOUBLE PRECISION RCOND
       DOUBLE PRECISION LOW(*), D(*), UP(*), LOWF(*), DF(*), UPF1(*), UPF2(*),
       B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)

       SUBROUTINE DGTSVX_64(FACT, TRANSA, N, NRHS, LOW, D, UP, LOWF,
	     DF, UPF1, UPF2, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR,
	     WORK, WORK2, INFO)

       CHARACTER * 1 FACT, TRANSA
       INTEGER*8 N, NRHS, LDB, LDX, INFO
       INTEGER*8 IPIVOT(*), WORK2(*)
       DOUBLE PRECISION RCOND
       DOUBLE PRECISION LOW(*), D(*), UP(*), LOWF(*), DF(*), UPF1(*), UPF2(*),
       B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)

   F95 INTERFACE
       SUBROUTINE GTSVX(FACT, [TRANSA], [N], [NRHS], LOW, D, UP, LOWF,
	      DF, UPF1, UPF2, IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR,
	      [WORK], [WORK2], [INFO])

       CHARACTER(LEN=1) :: FACT, TRANSA
       INTEGER :: N, NRHS, LDB, LDX, INFO
       INTEGER, DIMENSION(:) :: IPIVOT, WORK2
       REAL(8) :: RCOND
       REAL(8), DIMENSION(:) :: LOW, D, UP, LOWF, DF, UPF1, UPF2, FERR,	 BERR,
       WORK
       REAL(8), DIMENSION(:,:) :: B, X

       SUBROUTINE GTSVX_64(FACT, [TRANSA], [N], [NRHS], LOW, D, UP, LOWF,
	      DF, UPF1, UPF2, IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR,
	      [WORK], [WORK2], [INFO])

       CHARACTER(LEN=1) :: FACT, TRANSA
       INTEGER(8) :: N, NRHS, LDB, LDX, INFO
       INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2
       REAL(8) :: RCOND
       REAL(8),	 DIMENSION(:) :: LOW, D, UP, LOWF, DF, UPF1, UPF2, FERR, BERR,
       WORK
       REAL(8), DIMENSION(:,:) :: B, X

   C INTERFACE
       #include <sunperf.h>

       void dgtsvx(char fact, char transa, int n, int nrhs, double *low,  dou‐
		 ble  *d,  double *up, double *lowf, double *df, double *upf1,
		 double *upf2, int *ipivot, double *b, int ldb, double *x, int
		 ldx, double *rcond, double *ferr, double *berr, int *info);

       void  dgtsvx_64(char fact, char transa, long n, long nrhs, double *low,
		 double *d, double  *up,  double  *lowf,  double  *df,	double
		 *upf1,	 double *upf2, long *ipivot, double *b, long ldb, dou‐
		 ble *x, long ldx, double *rcond, double *ferr, double	*berr,
		 long *info);

PURPOSE
       dgtsvx uses the LU factorization to compute the solution to a real sys‐
       tem of linear equations A * X = B or A**T * X = B, where A is a	tridi‐
       agonal matrix of order N and X and B are N-by-NRHS matrices.

       Error  bounds  on  the  solution and a condition estimate are also pro‐
       vided.

       The following steps are performed:

       1. If FACT = 'N', the LU decomposition is used to factor the matrix A
	  as 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.

       2. If some U(i,i)=0, so that U is exactly singular, then the routine
	  returns with INFO = i. Otherwise, the factored form of A is used
	  to estimate the condition number of the matrix A.  If the
	  reciprocal of the condition number is less than machine precision,
	  INFO = N+1 is returned as a warning, but the routine still goes on
	  to solve for X and compute error bounds as described below.

       3. The system of equations is solved for X using the factored form
	  of A.

       4. Iterative refinement is applied to improve the computed solution
	  matrix and calculate error bounds and backward error estimates
	  for it.

ARGUMENTS
       FACT (input)
		 Specifies whether or not the factored form of A has been sup‐
		 plied	on  entry.   =	'F':  LOWF, DF, UPF1, UPF2, and IPIVOT
		 contain the factored form of A; LOW, D, UP, LOWF,  DF,	 UPF1,
		 UPF2  and  IPIVOT  will  not be modified.  = 'N':  The matrix
		 will be copied to LOWF, DF, and UPF1 and factored.

       TRANSA (input)
		 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)

		 TRANSA is defaulted to 'N' for F95 INTERFACE.

       N (input) The order of the matrix A.  N >= 0.

       NRHS (input)
		 The number of right hand sides, i.e., the number  of  columns
		 of the matrix B.  NRHS >= 0.

       LOW (input)
		 The (n-1) subdiagonal elements of A.

       D (input) The n diagonal elements of A.

       UP (input/output)
		 The (n-1) superdiagonal elements of A.

       LOWF (input or output)
		 If  FACT  =  'F', then LOWF is an input argument and on entry
		 contains the (n-1) multipliers that define the matrix L  from
		 the LU factorization of A as computed by DGTTRF.

		 If  FACT  =  'N', then LOWF is an output argument and on exit
		 contains the (n-1) multipliers that define the matrix L  from
		 the LU factorization of A.

       DF (input or output)
		 If FACT = 'F', then DF is an input argument and on entry con‐
		 tains the n diagonal elements of the upper triangular	matrix
		 U from the LU factorization of A.

		 If FACT = 'N', then DF is an output argument and on exit con‐
		 tains the n diagonal elements of the upper triangular	matrix
		 U from the LU factorization of A.

       UPF1 (input or output)
		 If  FACT  =  'F', then UPF1 is an input argument and on entry
		 contains the (n-1) elements of the first superdiagonal of U.

		 If FACT = 'N', then UPF1 is an output argument	 and  on  exit
		 contains the (n-1) elements of the first superdiagonal of U.

       UPF2 (input or output)
		 If  FACT  =  'F', then UPF2 is an input argument and on entry
		 contains the (n-2) elements of the second superdiagonal of U.

		 If FACT = 'N', then UPF2 is an output argument	 and  on  exit
		 contains the (n-2) elements of the second superdiagonal of U.

       IPIVOT (input/output)
		 If  FACT = 'F', then IPIVOT is an input argument and on entry
		 contains the pivot indices from the LU factorization of A  as
		 computed by DGTTRF.

		 If  FACT = 'N', then IPIVOT is an output argument and on exit
		 contains the pivot indices from the LU	 factorization	of  A;
		 row  i	 of  the  matrix  was interchanged with row IPIVOT(i).
		 IPIVOT(i) will always be either i or i+1; IPIVOT(i) = i indi‐
		 cates a row interchange was not required.

       B (input) The N-by-NRHS right hand side matrix B.

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

       X (output)
		 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

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

       RCOND (output)
		 The estimate of the reciprocal condition number of the matrix
		 A.  If RCOND is less than the machine precision (in  particu‐
		 lar,  if RCOND = 0), the matrix is singular to working preci‐
		 sion.	This condition is indicated by a return code of INFO >
		 0.

       FERR (output)
		 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 esti‐
		 mated upper bound for the magnitude of the largest element in
		 (X(j)	-  XTRUE) divided by the magnitude of the largest ele‐
		 ment 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 (output)
		 The componentwise relative backward error  of	each  solution
		 vector	 X(j)  (i.e., the smallest relative change in any ele‐
		 ment of A or B that makes X(j) an exact solution).

       WORK (workspace)
		 dimension(3*N)

       WORK2 (workspace)
		 dimension(N)

       INFO (output)
		 = 0:  successful exit
		 < 0:  if INFO = -i, the i-th argument had an illegal value
		 > 0:  if INFO = i, and i is
		 <= N:	U(i,i) is exactly zero.	  The  factorization  has  not
		 been completed unless i = N, but the factor U is exactly sin‐
		 gular, so the solution and error bounds  could	 not  be  com‐
		 puted.	  RCOND = 0 is returned.  = N+1: U is nonsingular, but
		 RCOND is less than machine precision, meaning that the matrix
		 is singular to working precision.  Nevertheless, the solution
		 and error bounds are computed because there are a  number  of
		 situations  where  the computed solution can be more accurate
		 than the value of RCOND would suggest.

				  6 Mar 2009			    dgtsvx(3P)
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