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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2.2) -- 00004 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- 00005 * -- Jason Riedy of Univ. of California Berkeley. -- 00006 * -- June 2010 -- 00007 * 00008 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00009 * -- Univ. of California Berkeley and NAG Ltd. -- 00010 * 00011 IMPLICIT NONE 00012 * .. 00013 * .. Scalar Arguments .. 00014 INTEGER INFO, LDA, N 00015 REAL AMAX, SCOND 00016 CHARACTER UPLO 00017 * .. 00018 * .. Array Arguments .. 00019 REAL A( LDA, * ), S( * ), WORK( * ) 00020 * .. 00021 * 00022 * Purpose 00023 * ======= 00024 * 00025 * SSYEQUB computes row and column scalings intended to equilibrate a 00026 * symmetric matrix A and reduce its condition number 00027 * (with respect to the two-norm). S contains the scale factors, 00028 * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with 00029 * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This 00030 * choice of S puts the condition number of B within a factor N of the 00031 * smallest possible condition number over all possible diagonal 00032 * scalings. 00033 * 00034 * Arguments 00035 * ========= 00036 * 00037 * UPLO (input) CHARACTER*1 00038 * Specifies whether the details of the factorization are stored 00039 * as an upper or lower triangular matrix. 00040 * = 'U': Upper triangular, form is A = U*D*U**T; 00041 * = 'L': Lower triangular, form is A = L*D*L**T. 00042 * 00043 * N (input) INTEGER 00044 * The order of the matrix A. N >= 0. 00045 * 00046 * A (input) REAL array, dimension (LDA,N) 00047 * The N-by-N symmetric matrix whose scaling 00048 * factors are to be computed. Only the diagonal elements of A 00049 * are referenced. 00050 * 00051 * LDA (input) INTEGER 00052 * The leading dimension of the array A. LDA >= max(1,N). 00053 * 00054 * S (output) REAL array, dimension (N) 00055 * If INFO = 0, S contains the scale factors for A. 00056 * 00057 * SCOND (output) REAL 00058 * If INFO = 0, S contains the ratio of the smallest S(i) to 00059 * the largest S(i). If SCOND >= 0.1 and AMAX is neither too 00060 * large nor too small, it is not worth scaling by S. 00061 * 00062 * AMAX (output) REAL 00063 * Absolute value of largest matrix element. If AMAX is very 00064 * close to overflow or very close to underflow, the matrix 00065 * should be scaled. 00066 * 00067 * WORK (workspace) REAL array, dimension (3*N) 00068 * 00069 * INFO (output) INTEGER 00070 * = 0: successful exit 00071 * < 0: if INFO = -i, the i-th argument had an illegal value 00072 * > 0: if INFO = i, the i-th diagonal element is nonpositive. 00073 * 00074 * Further Details 00075 * ======= ======= 00076 * 00077 * Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization", 00078 * Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. 00079 * DOI 10.1023/B:NUMA.0000016606.32820.69 00080 * Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf 00081 * 00082 * ===================================================================== 00083 * 00084 * .. Parameters .. 00085 REAL ONE, ZERO 00086 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) 00087 INTEGER MAX_ITER 00088 PARAMETER ( MAX_ITER = 100 ) 00089 * .. 00090 * .. Local Scalars .. 00091 INTEGER I, J, ITER 00092 REAL AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE, 00093 $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ 00094 LOGICAL UP 00095 * .. 00096 * .. External Functions .. 00097 REAL SLAMCH 00098 LOGICAL LSAME 00099 EXTERNAL LSAME, SLAMCH 00100 * .. 00101 * .. External Subroutines .. 00102 EXTERNAL SLASSQ 00103 * .. 00104 * .. Intrinsic Functions .. 00105 INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT 00106 * .. 00107 * .. Executable Statements .. 00108 * 00109 * Test input parameters. 00110 * 00111 INFO = 0 00112 IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN 00113 INFO = -1 00114 ELSE IF ( N .LT. 0 ) THEN 00115 INFO = -2 00116 ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN 00117 INFO = -4 00118 END IF 00119 IF ( INFO .NE. 0 ) THEN 00120 CALL XERBLA( 'SSYEQUB', -INFO ) 00121 RETURN 00122 END IF 00123 00124 UP = LSAME( UPLO, 'U' ) 00125 AMAX = ZERO 00126 * 00127 * Quick return if possible. 00128 * 00129 IF ( N .EQ. 0 ) THEN 00130 SCOND = ONE 00131 RETURN 00132 END IF 00133 00134 DO I = 1, N 00135 S( I ) = ZERO 00136 END DO 00137 00138 AMAX = ZERO 00139 IF ( UP ) THEN 00140 DO J = 1, N 00141 DO I = 1, J-1 00142 S( I ) = MAX( S( I ), ABS( A( I, J ) ) ) 00143 S( J ) = MAX( S( J ), ABS( A( I, J ) ) ) 00144 AMAX = MAX( AMAX, ABS( A(I, J) ) ) 00145 END DO 00146 S( J ) = MAX( S( J ), ABS( A( J, J ) ) ) 00147 AMAX = MAX( AMAX, ABS( A( J, J ) ) ) 00148 END DO 00149 ELSE 00150 DO J = 1, N 00151 S( J ) = MAX( S( J ), ABS( A( J, J ) ) ) 00152 AMAX = MAX( AMAX, ABS( A( J, J ) ) ) 00153 DO I = J+1, N 00154 S( I ) = MAX( S( I ), ABS( A( I, J ) ) ) 00155 S( J ) = MAX( S( J ), ABS( A( I, J ) ) ) 00156 AMAX = MAX( AMAX, ABS( A( I, J ) ) ) 00157 END DO 00158 END DO 00159 END IF 00160 DO J = 1, N 00161 S( J ) = 1.0 / S( J ) 00162 END DO 00163 00164 TOL = ONE / SQRT(2.0E0 * N) 00165 00166 DO ITER = 1, MAX_ITER 00167 SCALE = 0.0 00168 SUMSQ = 0.0 00169 * BETA = |A|S 00170 DO I = 1, N 00171 WORK(I) = ZERO 00172 END DO 00173 IF ( UP ) THEN 00174 DO J = 1, N 00175 DO I = 1, J-1 00176 T = ABS( A( I, J ) ) 00177 WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J ) 00178 WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I ) 00179 END DO 00180 WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J ) 00181 END DO 00182 ELSE 00183 DO J = 1, N 00184 WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J ) 00185 DO I = J+1, N 00186 T = ABS( A( I, J ) ) 00187 WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J ) 00188 WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I ) 00189 END DO 00190 END DO 00191 END IF 00192 00193 * avg = s^T beta / n 00194 AVG = 0.0 00195 DO I = 1, N 00196 AVG = AVG + S( I )*WORK( I ) 00197 END DO 00198 AVG = AVG / N 00199 00200 STD = 0.0 00201 DO I = 2*N+1, 3*N 00202 WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG 00203 END DO 00204 CALL SLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ ) 00205 STD = SCALE * SQRT( SUMSQ / N ) 00206 00207 IF ( STD .LT. TOL * AVG ) GOTO 999 00208 00209 DO I = 1, N 00210 T = ABS( A( I, I ) ) 00211 SI = S( I ) 00212 C2 = ( N-1 ) * T 00213 C1 = ( N-2 ) * ( WORK( I ) - T*SI ) 00214 C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG 00215 D = C1*C1 - 4*C0*C2 00216 00217 IF ( D .LE. 0 ) THEN 00218 INFO = -1 00219 RETURN 00220 END IF 00221 SI = -2*C0 / ( C1 + SQRT( D ) ) 00222 00223 D = SI - S( I ) 00224 U = ZERO 00225 IF ( UP ) THEN 00226 DO J = 1, I 00227 T = ABS( A( J, I ) ) 00228 U = U + S( J )*T 00229 WORK( J ) = WORK( J ) + D*T 00230 END DO 00231 DO J = I+1,N 00232 T = ABS( A( I, J ) ) 00233 U = U + S( J )*T 00234 WORK( J ) = WORK( J ) + D*T 00235 END DO 00236 ELSE 00237 DO J = 1, I 00238 T = ABS( A( I, J ) ) 00239 U = U + S( J )*T 00240 WORK( J ) = WORK( J ) + D*T 00241 END DO 00242 DO J = I+1,N 00243 T = ABS( A( J, I ) ) 00244 U = U + S( J )*T 00245 WORK( J ) = WORK( J ) + D*T 00246 END DO 00247 END IF 00248 00249 AVG = AVG + ( U + WORK( I ) ) * D / N 00250 S( I ) = SI 00251 00252 END DO 00253 00254 END DO 00255 00256 999 CONTINUE 00257 00258 SMLNUM = SLAMCH( 'SAFEMIN' ) 00259 BIGNUM = ONE / SMLNUM 00260 SMIN = BIGNUM 00261 SMAX = ZERO 00262 T = ONE / SQRT(AVG) 00263 BASE = SLAMCH( 'B' ) 00264 U = ONE / LOG( BASE ) 00265 DO I = 1, N 00266 S( I ) = BASE ** INT( U * LOG( S( I ) * T ) ) 00267 SMIN = MIN( SMIN, S( I ) ) 00268 SMAX = MAX( SMAX, S( I ) ) 00269 END DO 00270 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00271 * 00272 END
1.7.3 
