SUBROUTINE DGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX, $ BETA, Y, INCY ) * .. Scalar Arguments .. DOUBLE PRECISION ALPHA, BETA INTEGER INCX, INCY, LDA, M, N CHARACTER*1 TRANS * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) * .. * * Purpose * ======= * * DGEMV performs one of the matrix-vector operations * * y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y, * * where alpha and beta are scalars, x and y are vectors and A is an * m by n matrix. * * Parameters * ========== * * TRANS - CHARACTER*1. * On entry, TRANS specifies the operation to be performed as * follows: * * TRANS = 'N' or 'n' y := alpha*A*x + beta*y. * * TRANS = 'T' or 't' y := alpha*A'*x + beta*y. * * TRANS = 'C' or 'c' y := alpha*A'*x + beta*y. * * Unchanged on exit. * * M - INTEGER. * On entry, M specifies the number of rows of the matrix A. * M must be at least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of the matrix A. * N must be at least zero. * Unchanged on exit. * * ALPHA - DOUBLE PRECISION. * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). * Before entry, the leading m by n part of the array A must * contain the matrix of coefficients. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, m ). * Unchanged on exit. * * X - DOUBLE PRECISION array of DIMENSION at least * ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' * and at least * ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. * Before entry, the incremented array X must contain the * vector x. * Unchanged on exit. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * BETA - DOUBLE PRECISION. * On entry, BETA specifies the scalar beta. When BETA is * supplied as zero then Y need not be set on input. * Unchanged on exit. * * Y - DOUBLE PRECISION array of DIMENSION at least * ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' * and at least * ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. * Before entry with BETA non-zero, the incremented array Y * must contain the vector y. On exit, Y is overwritten by the * updated vector y. * * INCY - INTEGER. * On entry, INCY specifies the increment for the elements of * Y. INCY must not be zero. * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( TRANS, 'N' ).AND. $ .NOT.LSAME( TRANS, 'T' ).AND. $ .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 1 ELSE IF( M.LT.0 )THEN INFO = 2 ELSE IF( N.LT.0 )THEN INFO = 3 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 ELSE IF( INCY.EQ.0 )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DGEMV ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. $ ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) $ RETURN * * Set LENX and LENY, the lengths of the vectors x and y, and set * up the start points in X and Y. * IF( LSAME( TRANS, 'N' ) )THEN LENX = N LENY = M ELSE LENX = M LENY = N END IF IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( LENX - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( LENY - 1 )*INCY END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through A. * * First form y := beta*y. * IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, LENY Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, LENY Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, LENY Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, LENY Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) $ RETURN IF( LSAME( TRANS, 'N' ) )THEN * * Form y := alpha*A*x + y. * JX = KX IF( INCY.EQ.1 )THEN DO 60, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) DO 50, I = 1, M Y( I ) = Y( I ) + TEMP*A( I, J ) 50 CONTINUE END IF JX = JX + INCX 60 CONTINUE ELSE DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IY = KY DO 70, I = 1, M Y( IY ) = Y( IY ) + TEMP*A( I, J ) IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF ELSE * * Form y := alpha*A'*x + y. * JY = KY IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = ZERO DO 90, I = 1, M TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 100 CONTINUE ELSE DO 120, J = 1, N TEMP = ZERO IX = KX DO 110, I = 1, M TEMP = TEMP + A( I, J )*X( IX ) IX = IX + INCX 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 120 CONTINUE END IF END IF * RETURN * * End of DGEMV . * END SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) * * -- LAPACK driver routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * March 31, 1993 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) * .. * * Purpose * ======= * * DPOSV computes the solution to a real system of linear equations * A * X = B, * where A is an N-by-N symmetric positive definite matrix and X and B * are N-by-NRHS matrices. * * The Cholesky decomposition is used to factor A as * A = U**T* U, if UPLO = 'U', or * A = L * L**T, if UPLO = 'L', * where U is an upper triangular matrix and L is a lower triangular * matrix. The factored form of A is then used to solve the system of * equations A * X = B. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The number of linear equations, i.e., the order of the * matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the symmetric matrix A. If UPLO = 'U', the leading * N-by-N upper triangular part of A contains the upper * triangular part of the matrix A, and the strictly lower * triangular part of A is not referenced. If UPLO = 'L', the * leading N-by-N lower triangular part of A contains the lower * triangular part of the matrix A, and the strictly upper * triangular part of A is not referenced. * * On exit, if INFO = 0, the factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) * On entry, the N-by-NRHS right hand side matrix B. * On exit, if INFO = 0, the N-by-NRHS solution matrix X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the leading minor of order i of A is not * positive definite, so the factorization could not be * completed, and the solution has not been computed. * * ===================================================================== * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DPOTRF, DPOTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOSV ', -INFO ) RETURN END IF * * Compute the Cholesky factorization A = U'*U or A = L*L'. * CALL DPOTRF( UPLO, N, A, LDA, INFO ) IF( INFO.EQ.0 ) THEN * * Solve the system A*X = B, overwriting B with X. * CALL DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) * END IF RETURN * * End of DPOSV * END SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO ) * * -- LAPACK routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * February 29, 1992 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) * .. * * Purpose * ======= * * DPOTF2 computes the Cholesky factorization of a real symmetric * positive definite matrix A. * * The factorization has the form * A = U' * U , if UPLO = 'U', or * A = L * L', if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular. * * This is the unblocked version of the algorithm, calling Level 2 BLAS. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * symmetric matrix A is stored. * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the symmetric matrix A. If UPLO = 'U', the leading * n by n upper triangular part of A contains the upper * triangular part of the matrix A, and the strictly lower * triangular part of A is not referenced. If UPLO = 'L', the * leading n by n lower triangular part of A contains the lower * triangular part of the matrix A, and the strictly upper * triangular part of A is not referenced. * * On exit, if INFO = 0, the factor U or L from the Cholesky * factorization A = U'*U or A = L*L'. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -k, the k-th argument had an illegal value * > 0: if INFO = k, the leading minor of order k is not * positive definite, and the factorization could not be * completed. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER J DOUBLE PRECISION AJJ * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT * .. * .. External Subroutines .. EXTERNAL DGEMV, DSCAL, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTF2', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * IF( UPPER ) THEN * * Compute the Cholesky factorization A = U'*U. * DO 10 J = 1, N * * Compute U(J,J) and test for non-positive-definiteness. * AJJ = A( J, J ) - DDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 ) IF( AJJ.LE.ZERO ) THEN A( J, J ) = AJJ GO TO 30 END IF AJJ = SQRT( AJJ ) A( J, J ) = AJJ * * Compute elements J+1:N of row J. * IF( J.LT.N ) THEN CALL DGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ), $ LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA ) CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA ) END IF 10 CONTINUE ELSE * * Compute the Cholesky factorization A = L*L'. * DO 20 J = 1, N * * Compute L(J,J) and test for non-positive-definiteness. * AJJ = A( J, J ) - DDOT( J-1, A( J, 1 ), LDA, A( J, 1 ), $ LDA ) IF( AJJ.LE.ZERO ) THEN A( J, J ) = AJJ GO TO 30 END IF AJJ = SQRT( AJJ ) A( J, J ) = AJJ * * Compute elements J+1:N of column J. * IF( J.LT.N ) THEN CALL DGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1, 1 ), $ LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 ) CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 ) END IF 20 CONTINUE END IF GO TO 40 * 30 CONTINUE INFO = J * 40 CONTINUE RETURN * * End of DPOTF2 * END SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO ) * * -- LAPACK routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * March 31, 1993 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) * .. * * Purpose * ======= * * DPOTRF computes the Cholesky factorization of a real symmetric * positive definite matrix A. * * The factorization has the form * A = U**T * U, if UPLO = 'U', or * A = L * L**T, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular. * * This is the block version of the algorithm, calling Level 3 BLAS. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the symmetric matrix A. If UPLO = 'U', the leading * N-by-N upper triangular part of A contains the upper * triangular part of the matrix A, and the strictly lower * triangular part of A is not referenced. If UPLO = 'L', the * leading N-by-N lower triangular part of A contains the lower * triangular part of the matrix A, and the strictly upper * triangular part of A is not referenced. * * On exit, if INFO = 0, the factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the leading minor of order i is not * positive definite, and the factorization could not be * completed. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER J, JB, NB * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV * .. * .. External Subroutines .. EXTERNAL DGEMM, DPOTF2, DSYRK, DTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTRF', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Determine the block size for this environment. * NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 ) IF( NB.LE.1 .OR. NB.GE.N ) THEN * * Use unblocked code. * CALL DPOTF2( UPLO, N, A, LDA, INFO ) ELSE * * Use blocked code. * IF( UPPER ) THEN * * Compute the Cholesky factorization A = U'*U. * DO 10 J = 1, N, NB * * Update and factorize the current diagonal block and test * for non-positive-definiteness. * JB = MIN( NB, N-J+1 ) CALL DSYRK( 'Upper', 'Transpose', JB, J-1, -ONE, $ A( 1, J ), LDA, ONE, A( J, J ), LDA ) CALL DPOTF2( 'Upper', JB, A( J, J ), LDA, INFO ) IF( INFO.NE.0 ) $ GO TO 30 IF( J+JB.LE.N ) THEN * * Compute the current block row. * CALL DGEMM( 'Transpose', 'No transpose', JB, N-J-JB+1, $ J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ), $ LDA, ONE, A( J, J+JB ), LDA ) CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', $ JB, N-J-JB+1, ONE, A( J, J ), LDA, $ A( J, J+JB ), LDA ) END IF 10 CONTINUE * ELSE * * Compute the Cholesky factorization A = L*L'. * DO 20 J = 1, N, NB * * Update and factorize the current diagonal block and test * for non-positive-definiteness. * JB = MIN( NB, N-J+1 ) CALL DSYRK( 'Lower', 'No transpose', JB, J-1, -ONE, $ A( J, 1 ), LDA, ONE, A( J, J ), LDA ) CALL DPOTF2( 'Lower', JB, A( J, J ), LDA, INFO ) IF( INFO.NE.0 ) $ GO TO 30 IF( J+JB.LE.N ) THEN * * Compute the current block column. * CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, $ J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ), $ LDA, ONE, A( J+JB, J ), LDA ) CALL DTRSM( 'Right', 'Lower', 'Transpose', 'Non-unit', $ N-J-JB+1, JB, ONE, A( J, J ), LDA, $ A( J+JB, J ), LDA ) END IF 20 CONTINUE END IF END IF GO TO 40 * 30 CONTINUE INFO = INFO + J - 1 * 40 CONTINUE RETURN * * End of DPOTRF * END INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, $ N4 ) * * -- LAPACK auxiliary routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * September 30, 1994 * * .. Scalar Arguments .. CHARACTER*( * ) NAME, OPTS INTEGER ISPEC, N1, N2, N3, N4 * .. * * Purpose * ======= * * ILAENV is called from the LAPACK routines to choose problem-dependent * parameters for the local environment. See ISPEC for a description of * the parameters. * * This version provides a set of parameters which should give good, * but not optimal, performance on many of the currently available * computers. Users are encouraged to modify this subroutine to set * the tuning parameters for their particular machine using the option * and problem size information in the arguments. * * This routine will not function correctly if it is converted to all * lower case. Converting it to all upper case is allowed. * * Arguments * ========= * * ISPEC (input) INTEGER * Specifies the parameter to be returned as the value of * ILAENV. * = 1: the optimal blocksize; if this value is 1, an unblocked * algorithm will give the best performance. * = 2: the minimum block size for which the block routine * should be used; if the usable block size is less than * this value, an unblocked routine should be used. * = 3: the crossover point (in a block routine, for N less * than this value, an unblocked routine should be used) * = 4: the number of shifts, used in the nonsymmetric * eigenvalue routines * = 5: the minimum column dimension for blocking to be used; * rectangular blocks must have dimension at least k by m, * where k is given by ILAENV(2,...) and m by ILAENV(5,...) * = 6: the crossover point for the SVD (when reducing an m by n * matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds * this value, a QR factorization is used first to reduce * the matrix to a triangular form.) * = 7: the number of processors * = 8: the crossover point for the multishift QR and QZ methods * for nonsymmetric eigenvalue problems. * * NAME (input) CHARACTER*(*) * The name of the calling subroutine, in either upper case or * lower case. * * OPTS (input) CHARACTER*(*) * The character options to the subroutine NAME, concatenated * into a single character string. For example, UPLO = 'U', * TRANS = 'T', and DIAG = 'N' for a triangular routine would * be specified as OPTS = 'UTN'. * * N1 (input) INTEGER * N2 (input) INTEGER * N3 (input) INTEGER * N4 (input) INTEGER * Problem dimensions for the subroutine NAME; these may not all * be required. * * (ILAENV) (output) INTEGER * >= 0: the value of the parameter specified by ISPEC * < 0: if ILAENV = -k, the k-th argument had an illegal value. * * Further Details * =============== * * The following conventions have been used when calling ILAENV from the * LAPACK routines: * 1) OPTS is a concatenation of all of the character options to * subroutine NAME, in the same order that they appear in the * argument list for NAME, even if they are not used in determining * the value of the parameter specified by ISPEC. * 2) The problem dimensions N1, N2, N3, N4 are specified in the order * that they appear in the argument list for NAME. N1 is used * first, N2 second, and so on, and unused problem dimensions are * passed a value of -1. * 3) The parameter value returned by ILAENV is checked for validity in * the calling subroutine. For example, ILAENV is used to retrieve * the optimal blocksize for STRTRI as follows: * * NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 ) * IF( NB.LE.1 ) NB = MAX( 1, N ) * * ===================================================================== * * .. Local Scalars .. LOGICAL CNAME, SNAME CHARACTER*1 C1 CHARACTER*2 C2, C4 CHARACTER*3 C3 CHARACTER*6 SUBNAM INTEGER I, IC, IZ, NB, NBMIN, NX * .. * .. Intrinsic Functions .. INTRINSIC CHAR, ICHAR, INT, MIN, REAL * .. * .. Executable Statements .. * GO TO ( 100, 100, 100, 400, 500, 600, 700, 800 ) ISPEC * * Invalid value for ISPEC * ILAENV = -1 RETURN * 100 CONTINUE * * Convert NAME to upper case if the first character is lower case. * ILAENV = 1 SUBNAM = NAME IC = ICHAR( SUBNAM( 1:1 ) ) IZ = ICHAR( 'Z' ) IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN * * ASCII character set * IF( IC.GE.97 .AND. IC.LE.122 ) THEN SUBNAM( 1:1 ) = CHAR( IC-32 ) DO 10 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( IC.GE.97 .AND. IC.LE.122 ) $ SUBNAM( I:I ) = CHAR( IC-32 ) 10 CONTINUE END IF * ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN * * EBCDIC character set * IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR. $ ( IC.GE.145 .AND. IC.LE.153 ) .OR. $ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN SUBNAM( 1:1 ) = CHAR( IC+64 ) DO 20 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR. $ ( IC.GE.145 .AND. IC.LE.153 ) .OR. $ ( IC.GE.162 .AND. IC.LE.169 ) ) $ SUBNAM( I:I ) = CHAR( IC+64 ) 20 CONTINUE END IF * ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN * * Prime machines: ASCII+128 * IF( IC.GE.225 .AND. IC.LE.250 ) THEN SUBNAM( 1:1 ) = CHAR( IC-32 ) DO 30 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( IC.GE.225 .AND. IC.LE.250 ) $ SUBNAM( I:I ) = CHAR( IC-32 ) 30 CONTINUE END IF END IF * C1 = SUBNAM( 1:1 ) SNAME = C1.EQ.'S' .OR. C1.EQ.'D' CNAME = C1.EQ.'C' .OR. C1.EQ.'Z' IF( .NOT.( CNAME .OR. SNAME ) ) $ RETURN C2 = SUBNAM( 2:3 ) C3 = SUBNAM( 4:6 ) C4 = C3( 2:3 ) * GO TO ( 110, 200, 300 ) ISPEC * 110 CONTINUE * * ISPEC = 1: block size * * In these examples, separate code is provided for setting NB for * real and complex. We assume that NB will take the same value in * single or double precision. * NB = 1 * IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. $ C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'PO' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN NB = 1 ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN NB = 64 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRF' ) THEN NB = 64 ELSE IF( C3.EQ.'TRD' ) THEN NB = 1 ELSE IF( C3.EQ.'GST' ) THEN NB = 64 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NB = 32 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NB = 32 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NB = 32 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NB = 32 END IF END IF ELSE IF( C2.EQ.'GB' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN IF( N4.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF ELSE IF( N4.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF END IF END IF ELSE IF( C2.EQ.'PB' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN IF( N2.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF ELSE IF( N2.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF END IF END IF ELSE IF( C2.EQ.'TR' ) THEN IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'LA' ) THEN IF( C3.EQ.'UUM' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN IF( C3.EQ.'EBZ' ) THEN NB = 1 END IF END IF ILAENV = NB RETURN * 200 CONTINUE * * ISPEC = 2: minimum block size * NBMIN = 2 IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. $ C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NBMIN = 8 ELSE NBMIN = 8 END IF ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN NBMIN = 2 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRD' ) THEN NBMIN = 2 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NBMIN = 2 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NBMIN = 2 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NBMIN = 2 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NBMIN = 2 END IF END IF END IF ILAENV = NBMIN RETURN * 300 CONTINUE * * ISPEC = 3: crossover point * NX = 0 IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. $ C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( SNAME .AND. C3.EQ.'TRD' ) THEN NX = 1 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRD' ) THEN NX = 1 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NX = 128 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. $ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. $ C4.EQ.'BR' ) THEN NX = 128 END IF END IF END IF ILAENV = NX RETURN * 400 CONTINUE * * ISPEC = 4: number of shifts (used by xHSEQR) * ILAENV = 6 RETURN * 500 CONTINUE * * ISPEC = 5: minimum column dimension (not used) * ILAENV = 2 RETURN * 600 CONTINUE * * ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD) * ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 ) RETURN * 700 CONTINUE * * ISPEC = 7: number of processors (not used) * ILAENV = 1 RETURN * 800 CONTINUE * * ISPEC = 8: crossover point for multishift (used by xHSEQR) * ILAENV = 50 RETURN * * End of ILAENV * END LOGICAL FUNCTION LSAME( CA, CB ) * * -- LAPACK auxiliary routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * September 30, 1994 * * .. Scalar Arguments .. CHARACTER CA, CB * .. * * Purpose * ======= * * LSAME returns .TRUE. if CA is the same letter as CB regardless of * case. * * Arguments * ========= * * CA (input) CHARACTER*1 * CB (input) CHARACTER*1 * CA and CB specify the single characters to be compared. * * ===================================================================== * * .. Intrinsic Functions .. INTRINSIC ICHAR * .. * .. Local Scalars .. INTEGER INTA, INTB, ZCODE * .. * .. Executable Statements .. * * Test if the characters are equal * LSAME = CA.EQ.CB IF( LSAME ) $ RETURN * * Now test for equivalence if both characters are alphabetic. * ZCODE = ICHAR( 'Z' ) * * Use 'Z' rather than 'A' so that ASCII can be detected on Prime * machines, on which ICHAR returns a value with bit 8 set. * ICHAR('A') on Prime machines returns 193 which is the same as * ICHAR('A') on an EBCDIC machine. * INTA = ICHAR( CA ) INTB = ICHAR( CB ) * IF( ZCODE.EQ.90 .OR. ZCODE.EQ.122 ) THEN * * ASCII is assumed - ZCODE is the ASCII code of either lower or * upper case 'Z'. * IF( INTA.GE.97 .AND. INTA.LE.122 ) INTA = INTA - 32 IF( INTB.GE.97 .AND. INTB.LE.122 ) INTB = INTB - 32 * ELSE IF( ZCODE.EQ.233 .OR. ZCODE.EQ.169 ) THEN * * EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or * upper case 'Z'. * IF( INTA.GE.129 .AND. INTA.LE.137 .OR. $ INTA.GE.145 .AND. INTA.LE.153 .OR. $ INTA.GE.162 .AND. INTA.LE.169 ) INTA = INTA + 64 IF( INTB.GE.129 .AND. INTB.LE.137 .OR. $ INTB.GE.145 .AND. INTB.LE.153 .OR. $ INTB.GE.162 .AND. INTB.LE.169 ) INTB = INTB + 64 * ELSE IF( ZCODE.EQ.218 .OR. ZCODE.EQ.250 ) THEN * * ASCII is assumed, on Prime machines - ZCODE is the ASCII code * plus 128 of either lower or upper case 'Z'. * IF( INTA.GE.225 .AND. INTA.LE.250 ) INTA = INTA - 32 IF( INTB.GE.225 .AND. INTB.LE.250 ) INTB = INTB - 32 END IF LSAME = INTA.EQ.INTB * * RETURN * * End of LSAME * END SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) * * -- LAPACK routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * March 31, 1993 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) * .. * * Purpose * ======= * * DPOTRS solves a system of linear equations A*X = B with a symmetric * positive definite matrix A using the Cholesky factorization * A = U**T*U or A = L*L**T computed by DPOTRF. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The triangular factor U or L from the Cholesky factorization * A = U**T*U or A = L*L**T, as computed by DPOTRF. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) * On entry, the right hand side matrix B. * On exit, the solution matrix X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTRS', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 .OR. NRHS.EQ.0 ) $ RETURN * IF( UPPER ) THEN * * Solve A*X = B where A = U'*U. * * Solve U'*X = B, overwriting B with X. * CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS, $ ONE, A, LDA, B, LDB ) * * Solve U*X = B, overwriting B with X. * CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N, $ NRHS, ONE, A, LDA, B, LDB ) ELSE * * Solve A*X = B where A = L*L'. * * Solve L*X = B, overwriting B with X. * CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N, $ NRHS, ONE, A, LDA, B, LDB ) * * Solve L'*X = B, overwriting B with X. * CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N, NRHS, $ ONE, A, LDA, B, LDB ) END IF * RETURN * * End of DPOTRS * END * END SUBROUTINE DSYR ( UPLO, N, ALPHA, X, INCX, A, LDA ) * .. Scalar Arguments .. DOUBLE PRECISION ALPHA INTEGER INCX, LDA, N CHARACTER*1 UPLO * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ) * .. * * Purpose * ======= * * DSYR performs the symmetric rank 1 operation * * A := alpha*x*x' + A, * * where alpha is a real scalar, x is an n element vector and A is an * n by n symmetric matrix. * * Parameters * ========== * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the upper or lower * triangular part of the array A is to be referenced as * follows: * * UPLO = 'U' or 'u' Only the upper triangular part of A * is to be referenced. * * UPLO = 'L' or 'l' Only the lower triangular part of A * is to be referenced. * * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the order of the matrix A. * N must be at least zero. * Unchanged on exit. * * ALPHA - DOUBLE PRECISION. * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * X - DOUBLE PRECISION array of dimension at least * ( 1 + ( n - 1 )*abs( INCX ) ). * Before entry, the incremented array X must contain the n * element vector x. * Unchanged on exit. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). * Before entry with UPLO = 'U' or 'u', the leading n by n * upper triangular part of the array A must contain the upper * triangular part of the symmetric matrix and the strictly * lower triangular part of A is not referenced. On exit, the * upper triangular part of the array A is overwritten by the * upper triangular part of the updated matrix. * Before entry with UPLO = 'L' or 'l', the leading n by n * lower triangular part of the array A must contain the lower * triangular part of the symmetric matrix and the strictly * upper triangular part of A is not referenced. On exit, the * lower triangular part of the array A is overwritten by the * lower triangular part of the updated matrix. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, n ). * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) * .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, KX * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. $ .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 7 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYR ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) $ RETURN * * Set the start point in X if the increment is not unity. * IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through the triangular part * of A. * IF( LSAME( UPLO, 'U' ) )THEN * * Form A when A is stored in upper triangle. * IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = ALPHA*X( J ) DO 10, I = 1, J A( I, J ) = A( I, J ) + X( I )*TEMP 10 CONTINUE END IF 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IX = KX DO 30, I = 1, J A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 30 CONTINUE END IF JX = JX + INCX 40 CONTINUE END IF ELSE * * Form A when A is stored in lower triangle. * IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = ALPHA*X( J ) DO 50, I = J, N A( I, J ) = A( I, J ) + X( I )*TEMP 50 CONTINUE END IF 60 CONTINUE ELSE JX = KX DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IX = JX DO 70, I = J, N A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF END IF * RETURN * * End of DSYR . * END SUBROUTINE DTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, $ INFO ) * * -- LAPACK routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * March 31, 1993 * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, LDA, LDB, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) * .. * * Purpose * ======= * * DTRTRS solves a triangular system of the form * * A * X = B or A**T * X = B, * * where A is a triangular matrix of order N, and B is an N-by-NRHS * matrix. A check is made to verify that A is nonsingular. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': A is upper triangular; * = 'L': A is lower triangular. * * TRANS (input) 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) * * DIAG (input) CHARACTER*1 * = 'N': A is non-unit triangular; * = 'U': A is unit triangular. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The triangular matrix A. If UPLO = 'U', the leading N-by-N * upper triangular part of the array A contains the upper * triangular matrix, and the strictly lower triangular part of * A is not referenced. If UPLO = 'L', the leading N-by-N lower * triangular part of the array A contains the lower triangular * matrix, and the strictly upper triangular part of A is not * referenced. If DIAG = 'U', the diagonal elements of A are * also not referenced and are assumed to be 1. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) * On entry, the right hand side matrix B. * On exit, if INFO = 0, the solution matrix X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the i-th diagonal element of A is zero, * indicating that the matrix is singular and the solutions * X have not been computed. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL NOUNIT * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 NOUNIT = LSAME( DIAG, 'N' ) IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT. $ LSAME( TRANS, 'T' ) .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRTRS', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Check for singularity. * IF( NOUNIT ) THEN DO 10 INFO = 1, N IF( A( INFO, INFO ).EQ.ZERO ) $ RETURN 10 CONTINUE END IF INFO = 0 * * Solve A * x = b or A' * x = b. * CALL DTRSM( 'Left', UPLO, TRANS, DIAG, N, NRHS, ONE, A, LDA, B, $ LDB ) * RETURN * * End of DTRTRS * END * SUBROUTINE XERBLA( SRNAME, INFO ) * * -- LAPACK auxiliary routine (version 2.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * September 30, 1994 * * .. Scalar Arguments .. CHARACTER*6 SRNAME INTEGER INFO * .. * * Purpose * ======= * * XERBLA is an error handler for the LAPACK routines. * It is called by an LAPACK routine if an input parameter has an * invalid value. A message is printed and execution stops. * * Installers may consider modifying the STOP statement in order to * call system-specific exception-handling facilities. * * Arguments * ========= * * SRNAME (input) CHARACTER*6 * The name of the routine which called XERBLA. * * INFO (input) INTEGER * The position of the invalid parameter in the parameter list * of the calling routine. * * ===================================================================== * * .. Executable Statements .. * WRITE( *, FMT = 9999 )SRNAME, INFO * STOP * 9999 FORMAT( ' ** On entry to ', A6, ' parameter number ', I2, ' had ', $ 'an illegal value' ) * * End of XERBLA * END SUBROUTINE DTRSM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, $ B, LDB ) * .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO, TRANSA, DIAG INTEGER M, N, LDA, LDB DOUBLE PRECISION ALPHA * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) * .. * * Purpose * ======= * * DTRSM solves one of the matrix equations * * op( A )*X = alpha*B, or X*op( A ) = alpha*B, * * where alpha is a scalar, X and B are m by n matrices, A is a unit, or * non-unit, upper or lower triangular matrix and op( A ) is one of * * op( A ) = A or op( A ) = A'. * * The matrix X is overwritten on B. * * Parameters * ========== * * SIDE - CHARACTER*1. * On entry, SIDE specifies whether op( A ) appears on the left * or right of X as follows: * * SIDE = 'L' or 'l' op( A )*X = alpha*B. * * SIDE = 'R' or 'r' X*op( A ) = alpha*B. * * Unchanged on exit. * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the matrix A is an upper or * lower triangular matrix as follows: * * UPLO = 'U' or 'u' A is an upper triangular matrix. * * UPLO = 'L' or 'l' A is a lower triangular matrix. * * Unchanged on exit. * * TRANSA - CHARACTER*1. * On entry, TRANSA specifies the form of op( A ) to be used in * the matrix multiplication as follows: * * TRANSA = 'N' or 'n' op( A ) = A. * * TRANSA = 'T' or 't' op( A ) = A'. * * TRANSA = 'C' or 'c' op( A ) = A'. * * Unchanged on exit. * * DIAG - CHARACTER*1. * On entry, DIAG specifies whether or not A is unit triangular * as follows: * * DIAG = 'U' or 'u' A is assumed to be unit triangular. * * DIAG = 'N' or 'n' A is not assumed to be unit * triangular. * * Unchanged on exit. * * M - INTEGER. * On entry, M specifies the number of rows of B. M must be at * least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of B. N must be * at least zero. * Unchanged on exit. * * ALPHA - DOUBLE PRECISION. * On entry, ALPHA specifies the scalar alpha. When alpha is * zero then A is not referenced and B need not be set before * entry. * Unchanged on exit. * * A - DOUBLE PRECISION array of DIMENSION ( LDA, k ), where k is m * when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. * Before entry with UPLO = 'U' or 'u', the leading k by k * upper triangular part of the array A must contain the upper * triangular matrix and the strictly lower triangular part of * A is not referenced. * Before entry with UPLO = 'L' or 'l', the leading k by k * lower triangular part of the array A must contain the lower * triangular matrix and the strictly upper triangular part of * A is not referenced. * Note that when DIAG = 'U' or 'u', the diagonal elements of * A are not referenced either, but are assumed to be unity. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. When SIDE = 'L' or 'l' then * LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' * then LDA must be at least max( 1, n ). * Unchanged on exit. * * B - DOUBLE PRECISION array of DIMENSION ( LDB, n ). * Before entry, the leading m by n part of the array B must * contain the right-hand side matrix B, and on exit is * overwritten by the solution matrix X. * * LDB - INTEGER. * On entry, LDB specifies the first dimension of B as declared * in the calling (sub) program. LDB must be at least * max( 1, m ). * Unchanged on exit. * * * Level 3 Blas routine. * * * -- Written on 8-February-1989. * Jack Dongarra, Argonne National Laboratory. * Iain Duff, AERE Harwell. * Jeremy Du Croz, Numerical Algorithms Group Ltd. * Sven Hammarling, Numerical Algorithms Group Ltd. * * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. Local Scalars .. LOGICAL LSIDE, NOUNIT, UPPER INTEGER I, INFO, J, K, NROWA DOUBLE PRECISION TEMP * .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Executable Statements .. * * Test the input parameters. * LSIDE = LSAME( SIDE , 'L' ) IF( LSIDE )THEN NROWA = M ELSE NROWA = N END IF NOUNIT = LSAME( DIAG , 'N' ) UPPER = LSAME( UPLO , 'U' ) * INFO = 0 IF( ( .NOT.LSIDE ).AND. $ ( .NOT.LSAME( SIDE , 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. $ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 2 ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND. $ ( .NOT.LSAME( TRANSA, 'T' ) ).AND. $ ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN INFO = 3 ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND. $ ( .NOT.LSAME( DIAG , 'N' ) ) )THEN INFO = 4 ELSE IF( M .LT.0 )THEN INFO = 5 ELSE IF( N .LT.0 )THEN INFO = 6 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTRSM ', INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) $ RETURN * * And when alpha.eq.zero. * IF( ALPHA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE RETURN END IF * * Start the operations. * IF( LSIDE )THEN IF( LSAME( TRANSA, 'N' ) )THEN * * Form B := alpha*inv( A )*B. * IF( UPPER )THEN DO 60, J = 1, N IF( ALPHA.NE.ONE )THEN DO 30, I = 1, M B( I, J ) = ALPHA*B( I, J ) 30 CONTINUE END IF DO 50, K = M, 1, -1 IF( B( K, J ).NE.ZERO )THEN IF( NOUNIT ) $ B( K, J ) = B( K, J )/A( K, K ) DO 40, I = 1, K - 1 B( I, J ) = B( I, J ) - B( K, J )*A( I, K ) 40 CONTINUE END IF 50 CONTINUE 60 CONTINUE ELSE DO 100, J = 1, N IF( ALPHA.NE.ONE )THEN DO 70, I = 1, M B( I, J ) = ALPHA*B( I, J ) 70 CONTINUE END IF DO 90 K = 1, M IF( B( K, J ).NE.ZERO )THEN IF( NOUNIT ) $ B( K, J ) = B( K, J )/A( K, K ) DO 80, I = K + 1, M B( I, J ) = B( I, J ) - B( K, J )*A( I, K ) 80 CONTINUE END IF 90 CONTINUE 100 CONTINUE END IF ELSE * * Form B := alpha*inv( A' )*B. * IF( UPPER )THEN DO 130, J = 1, N DO 120, I = 1, M TEMP = ALPHA*B( I, J ) DO 110, K = 1, I - 1 TEMP = TEMP - A( K, I )*B( K, J ) 110 CONTINUE IF( NOUNIT ) $ TEMP = TEMP/A( I, I ) B( I, J ) = TEMP 120 CONTINUE 130 CONTINUE ELSE DO 160, J = 1, N DO 150, I = M, 1, -1 TEMP = ALPHA*B( I, J ) DO 140, K = I + 1, M TEMP = TEMP - A( K, I )*B( K, J ) 140 CONTINUE IF( NOUNIT ) $ TEMP = TEMP/A( I, I ) B( I, J ) = TEMP 150 CONTINUE 160 CONTINUE END IF END IF ELSE IF( LSAME( TRANSA, 'N' ) )THEN * * Form B := alpha*B*inv( A ). * IF( UPPER )THEN DO 210, J = 1, N IF( ALPHA.NE.ONE )THEN DO 170, I = 1, M B( I, J ) = ALPHA*B( I, J ) 170 CONTINUE END IF DO 190, K = 1, J - 1 IF( A( K, J ).NE.ZERO )THEN DO 180, I = 1, M B( I, J ) = B( I, J ) - A( K, J )*B( I, K ) 180 CONTINUE END IF 190 CONTINUE IF( NOUNIT )THEN TEMP = ONE/A( J, J ) DO 200, I = 1, M B( I, J ) = TEMP*B( I, J ) 200 CONTINUE END IF 210 CONTINUE ELSE DO 260, J = N, 1, -1 IF( ALPHA.NE.ONE )THEN DO 220, I = 1, M B( I, J ) = ALPHA*B( I, J ) 220 CONTINUE END IF DO 240, K = J + 1, N IF( A( K, J ).NE.ZERO )THEN DO 230, I = 1, M B( I, J ) = B( I, J ) - A( K, J )*B( I, K ) 230 CONTINUE END IF 240 CONTINUE IF( NOUNIT )THEN TEMP = ONE/A( J, J ) DO 250, I = 1, M B( I, J ) = TEMP*B( I, J ) 250 CONTINUE END IF 260 CONTINUE END IF ELSE * * Form B := alpha*B*inv( A' ). * IF( UPPER )THEN DO 310, K = N, 1, -1 IF( NOUNIT )THEN TEMP = ONE/A( K, K ) DO 270, I = 1, M B( I, K ) = TEMP*B( I, K ) 270 CONTINUE END IF DO 290, J = 1, K - 1 IF( A( J, K ).NE.ZERO )THEN TEMP = A( J, K ) DO 280, I = 1, M B( I, J ) = B( I, J ) - TEMP*B( I, K ) 280 CONTINUE END IF 290 CONTINUE IF( ALPHA.NE.ONE )THEN DO 300, I = 1, M B( I, K ) = ALPHA*B( I, K ) 300 CONTINUE END IF 310 CONTINUE ELSE DO 360, K = 1, N IF( NOUNIT )THEN TEMP = ONE/A( K, K ) DO 320, I = 1, M B( I, K ) = TEMP*B( I, K ) 320 CONTINUE END IF DO 340, J = K + 1, N IF( A( J, K ).NE.ZERO )THEN TEMP = A( J, K ) DO 330, I = 1, M B( I, J ) = B( I, J ) - TEMP*B( I, K ) 330 CONTINUE END IF 340 CONTINUE IF( ALPHA.NE.ONE )THEN DO 350, I = 1, M B( I, K ) = ALPHA*B( I, K ) 350 CONTINUE END IF 360 CONTINUE END IF END IF END IF * RETURN * * End of DTRSM . * END *DECK DGEMM SUBROUTINE DGEMM (TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, $ BETA, C, LDC) C***BEGIN PROLOGUE DGEMM C***PURPOSE Perform one of the matrix-matrix operations. C***LIBRARY SLATEC (BLAS) C***CATEGORY D1B6 C***TYPE DOUBLE PRECISION (SGEMM-S, DGEMM-D, CGEMM-C) C***KEYWORDS LEVEL 3 BLAS, LINEAR ALGEBRA C***AUTHOR Dongarra, J., (ANL) C Duff, I., (AERE) C Du Croz, J., (NAG) C Hammarling, S. (NAG) C***DESCRIPTION C C DGEMM performs one of the matrix-matrix operations C C C := alpha*op( A )*op( B ) + beta*C, C C where op( X ) is one of C C op( X ) = X or op( X ) = X', C C alpha and beta are scalars, and A, B and C are matrices, with op( A ) C an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. C C Parameters C ========== C C TRANSA - CHARACTER*1. C On entry, TRANSA specifies the form of op( A ) to be used in C the matrix multiplication as follows: C C TRANSA = 'N' or 'n', op( A ) = A. C C TRANSA = 'T' or 't', op( A ) = A'. C C TRANSA = 'C' or 'c', op( A ) = A'. C C Unchanged on exit. C C TRANSB - CHARACTER*1. C On entry, TRANSB specifies the form of op( B ) to be used in C the matrix multiplication as follows: C C TRANSB = 'N' or 'n', op( B ) = B. C C TRANSB = 'T' or 't', op( B ) = B'. C C TRANSB = 'C' or 'c', op( B ) = B'. C C Unchanged on exit. C C M - INTEGER. C On entry, M specifies the number of rows of the matrix C op( A ) and of the matrix C. M must be at least zero. C Unchanged on exit. C C N - INTEGER. C On entry, N specifies the number of columns of the matrix C op( B ) and the number of columns of the matrix C. N must be C at least zero. C Unchanged on exit. C C K - INTEGER. C On entry, K specifies the number of columns of the matrix C op( A ) and the number of rows of the matrix op( B ). K must C be at least zero. C Unchanged on exit. C C ALPHA - DOUBLE PRECISION. C On entry, ALPHA specifies the scalar alpha. C Unchanged on exit. C C A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is C k when TRANSA = 'N' or 'n', and is m otherwise. C Before entry with TRANSA = 'N' or 'n', the leading m by k C part of the array A must contain the matrix A, otherwise C the leading k by m part of the array A must contain the C matrix A. C Unchanged on exit. C C LDA - INTEGER. C On entry, LDA specifies the first dimension of A as declared C in the calling (sub) program. When TRANSA = 'N' or 'n' then C LDA must be at least max( 1, m ), otherwise LDA must be at C least max( 1, k ). C Unchanged on exit. C C B - DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is C n when TRANSB = 'N' or 'n', and is k otherwise. C Before entry with TRANSB = 'N' or 'n', the leading k by n C part of the array B must contain the matrix B, otherwise C the leading n by k part of the array B must contain the C matrix B. C Unchanged on exit. C C LDB - INTEGER. C On entry, LDB specifies the first dimension of B as declared C in the calling (sub) program. When TRANSB = 'N' or 'n' then C LDB must be at least max( 1, k ), otherwise LDB must be at C least max( 1, n ). C Unchanged on exit. C C BETA - DOUBLE PRECISION. C On entry, BETA specifies the scalar beta. When BETA is C supplied as zero then C need not be set on input. C Unchanged on exit. C C C - DOUBLE PRECISION array of DIMENSION ( LDC, n ). C Before entry, the leading m by n part of the array C must C contain the matrix C, except when beta is zero, in which C case C need not be set on entry. C On exit, the array C is overwritten by the m by n matrix C ( alpha*op( A )*op( B ) + beta*C ). C C LDC - INTEGER. C On entry, LDC specifies the first dimension of C as declared C in the calling (sub) program. LDC must be at least C max( 1, m ). C Unchanged on exit. C C***REFERENCES Dongarra, J., Du Croz, J., Duff, I., and Hammarling, S. C A set of level 3 basic linear algebra subprograms. C ACM TOMS, Vol. 16, No. 1, pp. 1-17, March 1990. C***ROUTINES CALLED LSAME, XERBLA C***REVISION HISTORY (YYMMDD) C 890208 DATE WRITTEN C 910605 Modified to meet SLATEC prologue standards. Only comment C lines were modified. (BKS) C***END PROLOGUE DGEMM C .. Scalar Arguments .. CHARACTER*1 TRANSA, TRANSB INTEGER M, N, K, LDA, LDB, LDC DOUBLE PRECISION ALPHA, BETA C .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ) C .. External Functions .. LOGICAL LSAME EXTERNAL LSAME C .. External Subroutines .. EXTERNAL XERBLA C .. Intrinsic Functions .. INTRINSIC MAX C .. Local Scalars .. LOGICAL NOTA, NOTB INTEGER I, INFO, J, L, NCOLA, NROWA, NROWB DOUBLE PRECISION TEMP C .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) C***FIRST EXECUTABLE STATEMENT DGEMM C C Set NOTA and NOTB as true if A and B respectively are not C transposed and set NROWA, NCOLA and NROWB as the number of rows C and columns of A and the number of rows of B respectively. C NOTA = LSAME( TRANSA, 'N' ) NOTB = LSAME( TRANSB, 'N' ) IF( NOTA )THEN NROWA = M NCOLA = K ELSE NROWA = K NCOLA = M END IF IF( NOTB )THEN NROWB = K ELSE NROWB = N END IF C C Test the input parameters. C INFO = 0 IF( ( .NOT.NOTA ).AND. $ ( .NOT.LSAME( TRANSA, 'C' ) ).AND. $ ( .NOT.LSAME( TRANSA, 'T' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.NOTB ).AND. $ ( .NOT.LSAME( TRANSB, 'C' ) ).AND. $ ( .NOT.LSAME( TRANSB, 'T' ) ) )THEN INFO = 2 ELSE IF( M .LT.0 )THEN INFO = 3 ELSE IF( N .LT.0 )THEN INFO = 4 ELSE IF( K .LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 8 ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN INFO = 10 ELSE IF( LDC.LT.MAX( 1, M ) )THEN INFO = 13 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DGEMM ', INFO ) RETURN END IF C C Quick return if possible. C IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. $ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) $ RETURN C C And if alpha.eq.zero. C IF( ALPHA.EQ.ZERO )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, M C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF RETURN END IF C C Start the operations. C IF( NOTB )THEN IF( NOTA )THEN C C Form C := alpha*A*B + beta*C. C DO 90, J = 1, N IF( BETA.EQ.ZERO )THEN DO 50, I = 1, M C( I, J ) = ZERO 50 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 60, I = 1, M C( I, J ) = BETA*C( I, J ) 60 CONTINUE END IF DO 80, L = 1, K IF( B( L, J ).NE.ZERO )THEN TEMP = ALPHA*B( L, J ) DO 70, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 70 CONTINUE END IF 80 CONTINUE 90 CONTINUE ELSE C C Form C := alpha*A'*B + beta*C C DO 120, J = 1, N DO 110, I = 1, M TEMP = ZERO DO 100, L = 1, K TEMP = TEMP + A( L, I )*B( L, J ) 100 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 110 CONTINUE 120 CONTINUE END IF ELSE IF( NOTA )THEN C C Form C := alpha*A*B' + beta*C C DO 170, J = 1, N IF( BETA.EQ.ZERO )THEN DO 130, I = 1, M C( I, J ) = ZERO 130 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 140, I = 1, M C( I, J ) = BETA*C( I, J ) 140 CONTINUE END IF DO 160, L = 1, K IF( B( J, L ).NE.ZERO )THEN TEMP = ALPHA*B( J, L ) DO 150, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 150 CONTINUE END IF 160 CONTINUE 170 CONTINUE ELSE C C Form C := alpha*A'*B' + beta*C C DO 200, J = 1, N DO 190, I = 1, M TEMP = ZERO DO 180, L = 1, K TEMP = TEMP + A( L, I )*B( J, L ) 180 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 190 CONTINUE 200 CONTINUE END IF END IF C RETURN C C End of DGEMM . C END *DECK DSYRK SUBROUTINE DSYRK (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC) C***BEGIN PROLOGUE DSYRK C***PURPOSE Perform one of the symmetric rank k operations. C***LIBRARY SLATEC (BLAS) C***CATEGORY D1B6 C***TYPE DOUBLE PRECISION (SSYRK-S, DSYRK-D, CSYRK-C) C***KEYWORDS LEVEL 3 BLAS, LINEAR ALGEBRA C***AUTHOR Dongarra, J., (ANL) C Duff, I., (AERE) C Du Croz, J., (NAG) C Hammarling, S. (NAG) C***DESCRIPTION C C DSYRK performs one of the symmetric rank k operations C C C := alpha*A*A' + beta*C, C C or C C C := alpha*A'*A + beta*C, C C where alpha and beta are scalars, C is an n by n symmetric matrix C and A is an n by k matrix in the first case and a k by n matrix C in the second case. C C Parameters C ========== C C UPLO - CHARACTER*1. C On entry, UPLO specifies whether the upper or lower C triangular part of the array C is to be referenced as C follows: C C UPLO = 'U' or 'u' Only the upper triangular part of C C is to be referenced. C C UPLO = 'L' or 'l' Only the lower triangular part of C C is to be referenced. C C Unchanged on exit. C C TRANS - CHARACTER*1. C On entry, TRANS specifies the operation to be performed as C follows: C C TRANS = 'N' or 'n' C := alpha*A*A' + beta*C. C C TRANS = 'T' or 't' C := alpha*A'*A + beta*C. C C TRANS = 'C' or 'c' C := alpha*A'*A + beta*C. C C Unchanged on exit. C C N - INTEGER. C On entry, N specifies the order of the matrix C. N must be C at least zero. C Unchanged on exit. C C K - INTEGER. C On entry with TRANS = 'N' or 'n', K specifies the number C of columns of the matrix A, and on entry with C TRANS = 'T' or 't' or 'C' or 'c', K specifies the number C of rows of the matrix A. K must be at least zero. C Unchanged on exit. C C ALPHA - DOUBLE PRECISION. C On entry, ALPHA specifies the scalar alpha. C Unchanged on exit. C C A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is C k when TRANS = 'N' or 'n', and is n otherwise. C Before entry with TRANS = 'N' or 'n', the leading n by k C part of the array A must contain the matrix A, otherwise C the leading k by n part of the array A must contain the C matrix A. C Unchanged on exit. C C LDA - INTEGER. C On entry, LDA specifies the first dimension of A as declared C in the calling (sub) program. When TRANS = 'N' or 'n' C then LDA must be at least max( 1, n ), otherwise LDA must C be at least max( 1, k ). C Unchanged on exit. C C BETA - DOUBLE PRECISION. C On entry, BETA specifies the scalar beta. C Unchanged on exit. C C C - DOUBLE PRECISION array of DIMENSION ( LDC, n ). C Before entry with UPLO = 'U' or 'u', the leading n by n C upper triangular part of the array C must contain the upper C triangular part of the symmetric matrix and the strictly C lower triangular part of C is not referenced. On exit, the C upper triangular part of the array C is overwritten by the C upper triangular part of the updated matrix. C Before entry with UPLO = 'L' or 'l', the leading n by n C lower triangular part of the array C must contain the lower C triangular part of the symmetric matrix and the strictly C upper triangular part of C is not referenced. On exit, the C lower triangular part of the array C is overwritten by the C lower triangular part of the updated matrix. C C LDC - INTEGER. C On entry, LDC specifies the first dimension of C as declared C in the calling (sub) program. LDC must be at least C max( 1, n ). C Unchanged on exit. C C***REFERENCES Dongarra, J., Du Croz, J., Duff, I., and Hammarling, S. C A set of level 3 basic linear algebra subprograms. C ACM TOMS, Vol. 16, No. 1, pp. 1-17, March 1990. C***ROUTINES CALLED LSAME, XERBLA C***REVISION HISTORY (YYMMDD) C 890208 DATE WRITTEN C 910605 Modified to meet SLATEC prologue standards. Only comment C lines were modified. (BKS) C***END PROLOGUE DSYRK C .. Scalar Arguments .. CHARACTER*1 UPLO, TRANS INTEGER N, K, LDA, LDC DOUBLE PRECISION ALPHA, BETA C .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ) C .. External Functions .. LOGICAL LSAME EXTERNAL LSAME C .. External Subroutines .. EXTERNAL XERBLA C .. Intrinsic Functions .. INTRINSIC MAX C .. Local Scalars .. LOGICAL UPPER INTEGER I, INFO, J, L, NROWA DOUBLE PRECISION TEMP C .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) C***FIRST EXECUTABLE STATEMENT DSYRK C C Test the input parameters. C IF( LSAME( TRANS, 'N' ) )THEN NROWA = N ELSE NROWA = K END IF UPPER = LSAME( UPLO, 'U' ) C INFO = 0 IF( ( .NOT.UPPER ).AND. $ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND. $ ( .NOT.LSAME( TRANS, 'T' ) ).AND. $ ( .NOT.LSAME( TRANS, 'C' ) ) )THEN INFO = 2 ELSE IF( N .LT.0 )THEN INFO = 3 ELSE IF( K .LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 7 ELSE IF( LDC.LT.MAX( 1, N ) )THEN INFO = 10 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYRK ', INFO ) RETURN END IF C C Quick return if possible. C IF( ( N.EQ.0 ).OR. $ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) $ RETURN C C And when alpha.eq.zero. C IF( ALPHA.EQ.ZERO )THEN IF( UPPER )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, J C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, J C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( BETA.EQ.ZERO )THEN DO 60, J = 1, N DO 50, I = J, N C( I, J ) = ZERO 50 CONTINUE 60 CONTINUE ELSE DO 80, J = 1, N DO 70, I = J, N C( I, J ) = BETA*C( I, J ) 70 CONTINUE 80 CONTINUE END IF END IF RETURN END IF C C Start the operations. C IF( LSAME( TRANS, 'N' ) )THEN C C Form C := alpha*A*A' + beta*C. C IF( UPPER )THEN DO 130, J = 1, N IF( BETA.EQ.ZERO )THEN DO 90, I = 1, J C( I, J ) = ZERO 90 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 100, I = 1, J C( I, J ) = BETA*C( I, J ) 100 CONTINUE END IF DO 120, L = 1, K IF( A( J, L ).NE.ZERO )THEN TEMP = ALPHA*A( J, L ) DO 110, I = 1, J C( I, J ) = C( I, J ) + TEMP*A( I, L ) 110 CONTINUE END IF 120 CONTINUE 130 CONTINUE ELSE DO 180, J = 1, N IF( BETA.EQ.ZERO )THEN DO 140, I = J, N C( I, J ) = ZERO 140 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 150, I = J, N C( I, J ) = BETA*C( I, J ) 150 CONTINUE END IF DO 170, L = 1, K IF( A( J, L ).NE.ZERO )THEN TEMP = ALPHA*A( J, L ) DO 160, I = J, N C( I, J ) = C( I, J ) + TEMP*A( I, L ) 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE END IF ELSE C C Form C := alpha*A'*A + beta*C. C IF( UPPER )THEN DO 210, J = 1, N DO 200, I = 1, J TEMP = ZERO DO 190, L = 1, K TEMP = TEMP + A( L, I )*A( L, J ) 190 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 200 CONTINUE 210 CONTINUE ELSE DO 240, J = 1, N DO 230, I = J, N TEMP = ZERO DO 220, L = 1, K TEMP = TEMP + A( L, I )*A( L, J ) 220 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 230 CONTINUE 240 CONTINUE END IF END IF C RETURN C C End of DSYRK . C END