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1 HSL MI6 PAKAGE SPEIFIATION HSL SUMMARY This routine uses the BiGStab (Bionjugate Gradient Stabilized) method to solve the n n unsymmetric linear system Ax = b, optionally using preconditioning. If P L, PR are the preconditioning matrices, the routine actually solves the preconditioned system Ax = b, with A = PLAPR and b = PLb and recovers the solution x = PRx. If P L = I, preconditioning is said to be from the right, if P R = I, it is said to be from the left, and otherwise it is from both sides. Reverse communication is used for preconditioning operations and matrix-vector products of the form Az. ATTRIBUTES Version: Types: Real (single, double). alls: FD15, _OPY, _DOT, _NRM, _SAL, _AXPY. Language: Fortran 77. Original date: March 001. Origin: N.I.M. Gould and J.A. Scott, Rutherford Appleton Laboratory. Remark: This is a threadsafe version of MI06A and supersedes it. HOW TO USE THE PAKAGE.1 Argument lists and calling sequence There are two entries: (a) MI6I/ID sets default values for control parameters. It should normally be called once prior to calls to MI6A/AD. (b) MI6A/AD uses the Bionjugate Gradient Stabilized method to solve Ax = b, optionally using preconditioning. MI6A/AD uses reverse communication for preconditioning operations and matrix-vector products..1.1 To set default values for the control parameters The single precision version ALL MI6I(INTL,NTL,ISAVE,RSAVE) The double precision version ALL MI6ID(INTL,NTL,ISAVE,RSAVE) INTL is an INTEGER array of length 8 that need not be set by the user. On return it contains default values (see Section. for details). NTL is a REAL (DOUBLE PREISION in the D version) array of length 5 that need not be set by the user. On return it contains default values (see Section. for details). ISAVE is an INTEGER array of length 14 used by MI6 as private workspace and must not be altered by the user. RSAVE is a REAL (DOUBLE PREISION in the D version) array of length 9 used by MI6 as private workspace and must not be altered by the user. HSL 007 MI6 Version Documentation date: 1st August 007 1

2 MI6 HSL To solve Ax = b The single precision version ALL MI6A(IAT,N,W,LDW,LOY,LOZ,RESID,INTL,NTL,INFO,ISAVE,RSAVE) The double precision version ALL MI6AD(IAT,N,W,LDW,LOY,LOZ,RESID,INTL,NTL,INFO,ISAVE,RSAVE) IAT is an INTEGER variable. Prior to the first call to MI6A/AD, IAT must be set by the user to 0. On each exit, IAT indicates the action required by the user. Possible values of IAT and the action required are as follows: N W LDW 1 An error has occurred and the user must terminate the computation (see INFO(1)). 1 If INTL(4) = 0 (the default), convergence has been achieved and the user should terminate the computation. If INTL(4) is nonzero, the user may test for convergence. If the user does not wish to test for convergence (we do not recommend the user tests for convergence each time IAT = 1 is returned) or if convergence has not been achieved, the user must recall MI6A/AD without changing any of the arguments. The user must perform the matrix-vector product y := Az and recall MI6A/AD. The vectors y and z are held in columns LOY and LOZ of array W, respectively. The vector z must be unchanged by the user. 3 The user must perform the preconditioning operation y := PLPRz, (.1) where P L, PR are the preconditioning matrices, and recall MI6A/AD. The vectors y and z are held in columns LOY and LOZ of array W, respectively. The vector z must be unchanged by the user. is an INTEGER variable that must be set by the user to n, the order of the matrix A. This variable must be preserved by the user between calls to MI6A/AD. This argument is not altered by the routine. Restriction: N 1. is a REAL (DOUBLE PREISION in the D version) two-dimensional array with dimensions (LDW, 8). Prior to the first call, column 1 must hold the right-hand side vector b and, if INTL(5) is nonzero, column must hold the initial estimate of the solution vector x. On exit with IAT = 1, column 1 holds the current residual vector r = b Ax, and the current estimate of the solution x is held in column. On exit with IAT > 1, the user is required to calculate y in column LOY of W (see argument IAT). The remaining columns of W must not be altered by the user between calls to MI6A/AD. is an INTEGER variable that must be set by the user to the first dimension of the array W. This argument is not altered by the routine. Restriction: LDW N. LOY,LOZ are INTEGER variables that need not be set by the user. On exit with IAT > 1, they indicate which columns of W contain the vectors y and z (see argument IAT). These arguments must not be altered by the user between calls to MI6A/AD. RESID is a REAL (DOUBLE PREISION in the D version) variable that need not be set by the user. On exit with IAT = 1, RESID holds the -norm of the current residual vector b Ax, where x is the current estimate of the solution. INTL is an INTEGER array of length 8 that contains control parameters. Default values for the components may be set by a call to MI6I/ID. Details of the control parameters are given in Section.. This argument is not altered by the routine. NTL is a REAL (DOUBLE PREISION in the D version) array of length 5 that contains control parameters and must be (.) MI6 Version Documentation date: 1st August 007 HSL 007

3 HSL 007 MI6 set by the user. Default values for the components may be set by a call to MI6I/ID. Details of the control parameters are given in Section.. This argument is not altered by the routine. INFO is an INTEGER array of length 4 that need not be set by the user. It is used to hold information about the execution of the subroutine. On exit from MI6A/AD, a value for INFO(1) of zero indicates that the subroutine has performed successfully. For nonzero values, see Section.3. INFO() holds the number of iterations performed.. INFO(3) and INFO(4) are not currently used. ISAVE is an INTEGER array of length 14 used by MI6 as private workspace and must not be altered by the user. RSAVE is a REAL (DOUBLE PREISION in the D version) array of length 9 used by MI6 as private workspace and must not be altered by the user.. ontrol parameters The elements of the arrays INTL and NTL control the action of MI6A/AD. Default values may be set by calling MI6I/ID. INTL(1) is the stream number for error messages and has the default value 6. Printing of error messages is suppressed if INTL(1) 0. INTL() is the stream number for warning messages and has the default value 6. Printing of warning messages is suppressed if INTL() 0. INTL(3) controls whether the user wishes to use preconditioning. It has default value 0 and in this case no preconditioning is used. If INTL(3) is nonzero, the user will be expected to perform preconditioning. INTL(4) controls whether the convergence test offered by MI6A/AD is to be used. It has default value 0 and in this case the computed solution x is accepted if b Ax is less than or equal to max( b Ax *NTL(1), NTL()), where x is the initial estimate of the solution. If the user does not want to use this test for convergence, INTL(4) should be nonzero. In this case, the user may test for convergence when IAT = 1 is returned. INTL(5) controls whether the user wishes to supply an initial estimate of the solution vector x. It has default value T 0 and in this case x = (0, 0,..., 0) is used as the initial estimate. If the user wishes to supply an initial estimate, INTL(5) should be nonzero and the initial estimate placed in the first N entries of column of the array W prior to the first call to MI6A/AD. INTL(6) determines the maximum number of iterations allowed. It has default value 1 and in this case the maximum number of iterations allowed is n. If the user does not want the maximum to be n, INTL(6) should be set to the maximum number of iterations the user does wish to allow. Values of INTL(6) which are less than or equal to zero are treated as if they were the default 1. INTRL(7) and INTRL(8) are set to zero by MI6I/ID but not currently used by MI6A/AD. NTL(1) and NTL() are the convergence tolerances (see Section 4). NTL(1) has default value u, where u is the relative machine precision (that is, the smallest machine number such that 1 + u > 1), while NTL() has default value zero. If INTL(4) is nonzero, NTL(1) and NTL() are not accessed by MI6A/AD. NTL(3) is the breakdown tolerance (see Section 4). It has default u, where u is the relative machine precision. NTRL(4) and NTRL(5) are set to zero by MI6I/ID but not currently used by MI6A/AD..3 Error diagnostics If MI6A/AD returns with a negative value of INFO(1), an error has occurred; if MI6A/AD returns with a positive value of INFO(1), a warning has been issued. Error messages are output on unit INTL(1) and warnings on unit INTL(). Possible non-zero values of INFO(1) are given below. 1 Value of N out of range. N < 1. Immediate return with input parameters unchanged. HSL 007 MI6 Version Documentation date: 1st August 007 3

4 MI6 HSL 007 Value of LDW out of range. LDW < N. Immediate return with input parameters unchanged. 3 Algorithm has broken down (see Section 4). 4 The maximum number of iterations determined by the control parameter INTL(6) has been exceeded. +1 The user-supplied convergence tolerance NTL(1) lies outside the interval (u,1.0), where u is the machine precision. NTL(1) is reset to the default convergence tolerance u..4 Underflows The nature of the calculations performed in this subroutine means that underflows are likely to occur. It is quite safe to set numbers that underflow to zero, and action by the user may be required to ensure that this is done efficiently by the computing system in use. 3 GENERAL INFORMATION Use of common: None. Other routines called directly: MI6A/AD calls HSL routine FD15A/AD, and the BLAS kernels SNRM/DNRM, SOPY/DOPY, SAXPY/DAXPY, SSAL/DSAL, SDOT/DDOT. Input/output: Error messages are printed on unit INTL(1) and warnings on unit INTL(); see Section.3. Restrictions: N 1, LDW N. 4 METHOD The Bionjugate Gradient Stabilized method is due to van der Vorst (199). The algorithm used by MI6A/AD proceeds as follows: heck the input data for errors. Set INFO(1) and return with IAT = 1 if a fatal error is encountered. if INTL(5) is nonzero let x be the initial guess supplied by the user Return with IAT = for the user to compute Ax. T set x = (0, 0,..., 0) ompute the initial residual r = b Ax if INTL(6) is greater then zero ITMAX = INTL(6) ITMAX = N do i = 1,,..., ITMAX T (i 1) ρ i 1 = (r ) r if ρ i 1 < NTL(3)*N (i 1) if ρ i 1 < NTL(3)* r r the method has broken down. Set INFO(1) and return with IAT = 1. if i = 1 (1) (1) p = u 4 MI6 Version Documentation date: 1st August 007 HSL 007

5 HSL 007 MI6 end do Reference β i 1 = (ρ i 1/ ρ i )(α i 1/ ω i 1) (i 1) i 1 i 1 p = r + β i 1(p ω i 1v ) if INTL(3) is nonzero Return with IAT = 3 for the user to compute p = PLPRp. p = p Return with IAT = for the user to compute v = Ap. T α i = ρ i 1/ (r ) v (i 1) s = r α i v (i 1) x = x + α i p Set r = s if INTL(4) is zero, (k) if r max( r NTL(1),NTL()) convergence has been achieved. Return with IAT = 1. Return with IAT = 1 to allow the user to check for convergence. if INTL(3) is nonzero Return with IAT = 3 for the user to compute ŝ = PLPRs. ŝ = s Return with IAT = for the user to compute t = Aŝ. T T ω i = t s/ t t if ω < NTL(3)*N i if ω i < NTL(3)* s / t the method has broken down. Set INFO(1) and return with IAT = 1. x = x + ω i ŝ r = s ω i t if INTL(4) is zero, (k) if r max( r NTL(1),NTL()) convergence has been achieved. Return with IAT = 1. Return with IAT = 1 to allow the user to check for convergence. H. van der Vorst (199). Bi-GSTAB: a fast and smoothly converging variant of Bi-G for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. omput., 13, HSL 007 MI6 Version Documentation date: 1st August 007 5

6 MI6 HSL EXAMPLE OF USE The following program illustrates the calling sequence for MI6. Solve the linear system A x = b, where A is tridiagonal with superdiagonal values 1, subdiagonals -1 and diagonals, and where the inverse of the diagonal of A is used as a preconditioner.. Parameters.. INTEGER N,LDW PARAMETER (N=10,LDW=N) DOUBLE PREISION TWO,ONE,THREE PARAMETER (TWO=.0D+0,ONE=1.0D+0,THREE=3.0D+0).... Local Scalars.. DOUBLE PREISION RESID INTEGER I,IAT,LOY,LOZ.... Local Arrays.. DOUBLE PREISION NTL(5),W(LDW,8) INTEGER INTL(8),INFO(4) INTEGER ISAVE(14) DOUBLE PREISION RSAVE(9).... External Subroutines.. EXTERNAL MI6AD,MI6ID.. ALL MI6ID(INTL,NTL,ISAVE,RSAVE) Preconditioning is required INTL(3) = 1 Set right hand side, b W(1,1) = THREE DO 10 I =,N - 1 W(I,1) = TWO 10 ONTINUE W(N,1) = ONE Perform an iteration of the Bionjugate Gradient Stabilized method IAT = 0 0 ONTINUE ALL MI6AD(IAT,N,W,LDW,LOY,LOZ,RESID,INTL,NTL,INFO,ISAVE, + RSAVE) IF (IAT.LT.0) THEN WRITE (6,FMT=900) INFO(1) GO TO 50 END IF IF (IAT.EQ.) THEN Perform the matrix-vector product W(1,LOY) = TWO*W(1,LOZ) + W(,LOZ) DO 40 I =,N - 1 W(I,LOY) = -W(I-1,LOZ) + TWO*W(I,LOZ) + W(I+1,LOZ) 40 ONTINUE W(N,LOY) = -W(N-1,LOZ) + TWO*W(N,LOZ) GO TO 0 END IF 6 MI6 Version Documentation date: 1st August 007 HSL 007

7 HSL 007 MI6 IF (IAT.EQ.3) THEN Perform the preconditioning operation DO 30 I = 1,N W(I,LOY) = W(I,LOZ)/TWO 30 ONTINUE GO TO 0 END IF Solution found WRITE (6,FMT=9000) INFO(), (W(I,),I=1,N) IF (INFO(1).GT.0) WRITE (6,FMT=9010) INFO(1) 50 ONTINUE STOP 9000 FORMAT (I6,' iterations required by MI6 ',//' Solution = ', + / (1P,5D10.)) 9010 FORMAT (' Warning: INFO( 1 ) = ',I,' on exit ') 900 FORMAT (' Error return: INFO( 1 ) = ',I,' on exit ') END This produces the following output: 10 iterations required by MI6 Solution = 1.00D D D D D D D D D D+00 HSL 007 MI6 Version Documentation date: 1st August 007 7