SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N * .. * .. Array Arguments .. COMPLEX AB( LDAB, * ) * .. * * Purpose * ======= * * CPBSTF computes a split Cholesky factorization of a complex * Hermitian positive definite band matrix A. * * This routine is designed to be used in conjunction with CHBGST. * * The factorization has the form A = S**H*S where S is a band matrix * of the same bandwidth as A and the following structure: * * S = ( U ) * ( M L ) * * where U is upper triangular of order m = (n+kd)/2, and L is lower * triangular of order n-m. * * 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. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input/output) COMPLEX array, dimension (LDAB,N) * On entry, the upper or lower triangle of the Hermitian band * matrix A, stored in the first kd+1 rows of the array. The * j-th column of A is stored in the j-th column of the array AB * as follows: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). * * On exit, if INFO = 0, the factor S from the split Cholesky * factorization A = S**H*S. See Further Details. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD+1. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the factorization could not be completed, * because the updated element a(i,i) was negative; the * matrix A is not positive definite. * * Further Details * =============== * * The band storage scheme is illustrated by the following example, when * N = 7, KD = 2: * * S = ( s11 s12 s13 ) * ( s22 s23 s24 ) * ( s33 s34 ) * ( s44 ) * ( s53 s54 s55 ) * ( s64 s65 s66 ) * ( s75 s76 s77 ) * * If UPLO = 'U', the array AB holds: * * on entry: on exit: * * * * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75' * * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76' * a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 * * If UPLO = 'L', the array AB holds: * * on entry: on exit: * * a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 * a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * * a31 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * * * * Array elements marked * are not used by the routine; s12' denotes * conjg(s12); the diagonal elements of S are real. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER J, KLD, KM, M REAL AJJ * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL CHER, CLACGV, CSSCAL, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL, 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( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'CPBSTF', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * KLD = MAX( 1, LDAB-1 ) * * Set the splitting point m. * M = ( N+KD ) / 2 * IF( UPPER ) THEN * * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). * DO 10 J = N, M + 1, -1 * * Compute s(j,j) and test for non-positive-definiteness. * AJJ = REAL( AB( KD+1, J ) ) IF( AJJ.LE.ZERO ) THEN AB( KD+1, J ) = AJJ GO TO 50 END IF AJJ = SQRT( AJJ ) AB( KD+1, J ) = AJJ KM = MIN( J-1, KD ) * * Compute elements j-km:j-1 of the j-th column and update the * the leading submatrix within the band. * CALL CSSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) CALL CHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, \$ AB( KD+1, J-KM ), KLD ) 10 CONTINUE * * Factorize the updated submatrix A(1:m,1:m) as U**H*U. * DO 20 J = 1, M * * Compute s(j,j) and test for non-positive-definiteness. * AJJ = REAL( AB( KD+1, J ) ) IF( AJJ.LE.ZERO ) THEN AB( KD+1, J ) = AJJ GO TO 50 END IF AJJ = SQRT( AJJ ) AB( KD+1, J ) = AJJ KM = MIN( KD, M-J ) * * Compute elements j+1:j+km of the j-th row and update the * trailing submatrix within the band. * IF( KM.GT.0 ) THEN CALL CSSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) CALL CLACGV( KM, AB( KD, J+1 ), KLD ) CALL CHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, \$ AB( KD+1, J+1 ), KLD ) CALL CLACGV( KM, AB( KD, J+1 ), KLD ) END IF 20 CONTINUE ELSE * * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). * DO 30 J = N, M + 1, -1 * * Compute s(j,j) and test for non-positive-definiteness. * AJJ = REAL( AB( 1, J ) ) IF( AJJ.LE.ZERO ) THEN AB( 1, J ) = AJJ GO TO 50 END IF AJJ = SQRT( AJJ ) AB( 1, J ) = AJJ KM = MIN( J-1, KD ) * * Compute elements j-km:j-1 of the j-th row and update the * trailing submatrix within the band. * CALL CSSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) CALL CLACGV( KM, AB( KM+1, J-KM ), KLD ) CALL CHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, \$ AB( 1, J-KM ), KLD ) CALL CLACGV( KM, AB( KM+1, J-KM ), KLD ) 30 CONTINUE * * Factorize the updated submatrix A(1:m,1:m) as U**H*U. * DO 40 J = 1, M * * Compute s(j,j) and test for non-positive-definiteness. * AJJ = REAL( AB( 1, J ) ) IF( AJJ.LE.ZERO ) THEN AB( 1, J ) = AJJ GO TO 50 END IF AJJ = SQRT( AJJ ) AB( 1, J ) = AJJ KM = MIN( KD, M-J ) * * Compute elements j+1:j+km of the j-th column and update the * trailing submatrix within the band. * IF( KM.GT.0 ) THEN CALL CSSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) CALL CHER( 'Lower', KM, -ONE, AB( 2, J ), 1, \$ AB( 1, J+1 ), KLD ) END IF 40 CONTINUE END IF RETURN * 50 CONTINUE INFO = J RETURN * * End of CPBSTF * END