1: #include <petscsys.h>
2: #include <petscblaslapack.h>
6: static PetscErrorCode estsv(PetscInt n, PetscReal *r, PetscInt ldr, PetscReal *svmin, PetscReal *z) 7: {
8: PetscBLASInt blas1=1, blasn=n, blasnmi, blasj, blasldr = ldr;
9: PetscInt i,j;
10: PetscReal e,temp,w,wm,ynorm,znorm,s,sm;
13: for (i=0;i<n;i++) {
14: z[i]=0.0;
15: }
16: e = PetscAbs(r[0]);
17: if (e == 0.0) {
18: *svmin = 0.0;
19: z[0] = 1.0;
20: } else {
21: /* Solve R'*y = e */
22: for (i=0;i<n;i++) {
23: /* Scale y. The scaling factor (0.01) reduces the number of scalings */
24: if (z[i] >= 0.0) e =-PetscAbs(e);
25: else e = PetscAbs(e);
27: if (PetscAbs(e - z[i]) > PetscAbs(r[i + ldr*i])) {
28: temp = PetscMin(0.01,PetscAbs(r[i + ldr*i]))/PetscAbs(e-z[i]);
29: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, z, &blas1));
30: e = temp*e;
31: }
33: /* Determine the two possible choices of y[i] */
34: if (r[i + ldr*i] == 0.0) {
35: w = wm = 1.0;
36: } else {
37: w = (e - z[i]) / r[i + ldr*i];
38: wm = - (e + z[i]) / r[i + ldr*i];
39: }
41: /* Chose y[i] based on the predicted value of y[j] for j>i */
42: s = PetscAbs(e - z[i]);
43: sm = PetscAbs(e + z[i]);
44: for (j=i+1;j<n;j++) {
45: sm += PetscAbs(z[j] + wm * r[i + ldr*j]);
46: }
47: if (i < n-1) {
48: blasnmi = n-i-1;
49: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasnmi, &w, &r[i + ldr*(i+1)], &blasldr, &z[i+1], &blas1));
50: s += BLASasum_(&blasnmi, &z[i+1], &blas1);
51: }
52: if (s < sm) {
53: temp = wm - w;
54: w = wm;
55: if (i < n-1) {
56: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasnmi, &temp, &r[i + ldr*(i+1)], &blasldr, &z[i+1], &blas1));
57: }
58: }
59: z[i] = w;
60: }
62: ynorm = BLASnrm2_(&blasn, z, &blas1);
64: /* Solve R*z = y */
65: for (j=n-1; j>=0; j--) {
66: /* Scale z */
67: if (PetscAbs(z[j]) > PetscAbs(r[j + ldr*j])) {
68: temp = PetscMin(0.01, PetscAbs(r[j + ldr*j] / z[j]));
69: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, z, &blas1));
70: ynorm *=temp;
71: }
72: if (r[j + ldr*j] == 0) {
73: z[j] = 1.0;
74: } else {
75: z[j] = z[j] / r[j + ldr*j];
76: }
77: temp = -z[j];
78: blasj=j;
79: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasj,&temp,&r[0+ldr*j],&blas1,z,&blas1));
80: }
82: /* Compute svmin and normalize z */
83: znorm = 1.0 / BLASnrm2_(&blasn, z, &blas1);
84: *svmin = ynorm*znorm;
85: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &znorm, z, &blas1));
86: }
87: return(0);
88: }
90: /*
91: c ***********
92: c
93: c Subroutine dgqt
94: c
95: c Given an n by n symmetric matrix A, an n-vector b, and a
96: c positive number delta, this subroutine determines a vector
97: c x which approximately minimizes the quadratic function
98: c 99: c f(x) = (1/2)*x'*A*x + b'*x
100: c101: c subject to the Euclidean norm constraint102: c103: c norm(x) <= delta.
104: c105: c This subroutine computes an approximation x and a Lagrange106: c multiplier par such that either par is zero and107: c108: c norm(x) <= (1+rtol)*delta,
109: c110: c or par is positive and111: c112: c abs(norm(x) - delta) <= rtol*delta.
113: c
114: c If xsol is the solution to the problem, the approximation x
115: c satisfies116: c117: c f(x) <= ((1 - rtol)**2)*f(xsol)
118: c119: c The subroutine statement is120: c121: c subroutine dgqt(n,a,lda,b,delta,rtol,atol,itmax,122: c par,f,x,info,z,wa1,wa2)123: c
124: c where
125: c
126: c n is an integer variable.
127: c On entry n is the order of A.
128: c On exit n is unchanged.
129: c130: c a is a double precision array of dimension (lda,n).
131: c On entry the full upper triangle of a must contain the
132: c full upper triangle of the symmetric matrix A.
133: c On exit the array contains the matrix A.
134: c
135: c lda is an integer variable.
136: c On entry lda is the leading dimension of the array a.
137: c On exit lda is unchanged.
138: c
139: c b is an double precision array of dimension n.
140: c On entry b specifies the linear term in the quadratic.
141: c On exit b is unchanged.
142: c
143: c delta is a double precision variable.
144: c On entry delta is a bound on the Euclidean norm of x.
145: c On exit delta is unchanged.
146: c
147: c rtol is a double precision variable.
148: c On entry rtol is the relative accuracy desired in the
149: c solution. Convergence occurs if
150: c151: c f(x) <= ((1 - rtol)**2)*f(xsol)
152: c
153: c On exit rtol is unchanged.
154: c
155: c atol is a double precision variable.
156: c On entry atol is the absolute accuracy desired in the
157: c solution. Convergence occurs when
158: c159: c norm(x) <= (1 + rtol)*delta
160: c
161: c max(-f(x),-f(xsol)) <= atol
162: c
163: c On exit atol is unchanged.
164: c
165: c itmax is an integer variable.
166: c On entry itmax specifies the maximum number of iterations.
167: c On exit itmax is unchanged.
168: c
169: c par is a double precision variable.
170: c On entry par is an initial estimate of the Lagrange171: c multiplier for the constraint norm(x) <= delta.
172: c On exit par contains the final estimate of the multiplier.
173: c
174: c f is a double precision variable.
175: c On entry f need not be specified.
176: c On exit f is set to f(x) at the output x.
177: c
178: c x is a double precision array of dimension n.
179: c On entry x need not be specified.
180: c On exit x is set to the final estimate of the solution.
181: c
182: c info is an integer variable.
183: c On entry info need not be specified.
184: c On exit info is set as follows:
185: c
186: c info = 1 The function value f(x) has the relative
187: c accuracy specified by rtol.
188: c
189: c info = 2 The function value f(x) has the absolute
190: c accuracy specified by atol.
191: c
192: c info = 3 Rounding errors prevent further progress.
193: c On exit x is the best available approximation.
194: c
195: c info = 4 Failure to converge after itmax iterations.
196: c On exit x is the best available approximation.
197: c
198: c z is a double precision work array of dimension n.
199: c
200: c wa1 is a double precision work array of dimension n.
201: c
202: c wa2 is a double precision work array of dimension n.
203: c
204: c Subprograms called
205: c
206: c MINPACK-2 ...... destsv
207: c
208: c LAPACK ......... dpotrf
209: c
210: c Level 1 BLAS ... daxpy, dcopy, ddot, dnrm2, dscal
211: c
212: c Level 2 BLAS ... dtrmv, dtrsv
213: c
214: c MINPACK-2 Project. October 1993.
215: c Argonne National Laboratory and University of Minnesota.
216: c Brett M. Averick, Richard Carter, and Jorge J. More'
217: c
218: c ***********
219: */
222: PetscErrorCode gqt(PetscInt n, PetscReal *a, PetscInt lda, PetscReal *b,223: PetscReal delta, PetscReal rtol, PetscReal atol,224: PetscInt itmax, PetscReal *retpar, PetscReal *retf,225: PetscReal *x, PetscInt *retinfo, PetscInt *retits,226: PetscReal *z, PetscReal *wa1, PetscReal *wa2)227: {
229: PetscReal f=0.0,p001=0.001,p5=0.5,minusone=-1,delta2=delta*delta;
230: PetscInt iter, j, rednc,info;
231: PetscBLASInt indef;
232: PetscBLASInt blas1=1, blasn=n, iblas, blaslda = lda,blasldap1=lda+1,blasinfo;
233: PetscReal alpha, anorm, bnorm, parc, parf, parl, pars, par=*retpar,paru, prod, rxnorm, rznorm=0.0, temp, xnorm;
236: parf = 0.0;
237: xnorm = 0.0;
238: rxnorm = 0.0;
239: rednc = 0;
240: for (j=0; j<n; j++) {
241: x[j] = 0.0;
242: z[j] = 0.0;
243: }
245: /* Copy the diagonal and save A in its lower triangle */
246: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,a,&blasldap1, wa1, &blas1));
247: for (j=0;j<n-1;j++) {
248: iblas = n - j - 1;
249: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j + lda*(j+1)], &blaslda, &a[j+1 + lda*j], &blas1));
250: }
252: /* Calculate the l1-norm of A, the Gershgorin row sums, and the
253: l2-norm of b */
254: anorm = 0.0;
255: for (j=0;j<n;j++) {
256: wa2[j] = BLASasum_(&blasn, &a[0 + lda*j], &blas1);
257: CHKMEMQ;
258: anorm = PetscMax(anorm,wa2[j]);
259: }
260: for (j=0;j<n;j++) {
261: wa2[j] = wa2[j] - PetscAbs(wa1[j]);
262: }
263: bnorm = BLASnrm2_(&blasn,b,&blas1);
264: CHKMEMQ;
265: /* Calculate a lower bound, pars, for the domain of the problem.
266: Also calculate an upper bound, paru, and a lower bound, parl,
267: for the Lagrange multiplier. */
268: pars = parl = paru = -anorm;
269: for (j=0;j<n;j++) {
270: pars = PetscMax(pars, -wa1[j]);
271: parl = PetscMax(parl, wa1[j] + wa2[j]);
272: paru = PetscMax(paru, -wa1[j] + wa2[j]);
273: }
274: parl = PetscMax(bnorm/delta - parl,pars);
275: parl = PetscMax(0.0,parl);
276: paru = PetscMax(0.0, bnorm/delta + paru);
278: /* If the input par lies outside of the interval (parl, paru),
279: set par to the closer endpoint. */
281: par = PetscMax(par,parl);
282: par = PetscMin(par,paru);
284: /* Special case: parl == paru */
285: paru = PetscMax(paru, (1.0 + rtol)*parl);
287: /* Beginning of an iteration */
289: info = 0;
290: for (iter=1;iter<=itmax;iter++) {
291: /* Safeguard par */
292: if (par <= pars && paru > 0) {
293: par = PetscMax(p001, PetscSqrtScalar(parl/paru)) * paru;
294: }
296: /* Copy the lower triangle of A into its upper triangle and
297: compute A + par*I */
299: for (j=0;j<n-1;j++) {
300: iblas = n - j - 1;
301: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j+1 + j*lda], &blas1,&a[j + (j+1)*lda], &blaslda));
302: }
303: for (j=0;j<n;j++) {
304: a[j + j*lda] = wa1[j] + par;
305: }
307: /* Attempt the Cholesky factorization of A without referencing
308: the lower triangular part. */
309: PetscStackCallBLAS("LAPACKpotrf",LAPACKpotrf_("U",&blasn,a,&blaslda,&indef));
311: /* Case 1: A + par*I is pos. def. */
312: if (indef == 0) {
314: /* Compute an approximate solution x and save the
315: last value of par with A + par*I pos. def. */
317: parf = par;
318: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, b, &blas1, wa2, &blas1));
319: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&blasn,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
320: rxnorm = BLASnrm2_(&blasn, wa2, &blas1);
321: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","N","N",&blasn,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
322: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, wa2, &blas1, x, &blas1));
323: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &minusone, x, &blas1));
324: xnorm = BLASnrm2_(&blasn, x, &blas1);
325: CHKMEMQ;
327: /* Test for convergence */
328: if (PetscAbs(xnorm - delta) <= rtol*delta ||
329: (par == 0 && xnorm <= (1.0+rtol)*delta)) {
330: info = 1;
331: }
333: /* Compute a direction of negative curvature and use this
334: information to improve pars. */
336: iblas=blasn*blasn;
338: estsv(n,a,lda,&rznorm,z);
339: CHKMEMQ;
340: pars = PetscMax(pars, par-rznorm*rznorm);
342: /* Compute a negative curvature solution of the form
343: x + alpha*z, where norm(x+alpha*z)==delta */
345: rednc = 0;
346: if (xnorm < delta) {
347: /* Compute alpha */
348: prod = BLASdot_(&blasn, z, &blas1, x, &blas1) / delta;
349: temp = (delta - xnorm)*((delta + xnorm)/delta);
350: alpha = temp/(PetscAbs(prod) + PetscSqrtScalar(prod*prod + temp/delta));
351: if (prod >= 0) alpha = PetscAbs(alpha);
352: else alpha =-PetscAbs(alpha);
354: /* Test to decide if the negative curvature step
355: produces a larger reduction than with z=0 */
356: rznorm = PetscAbs(alpha) * rznorm;
357: if ((rznorm*rznorm + par*xnorm*xnorm)/(delta2) <= par) {
358: rednc = 1;
359: }
360: /* Test for convergence */
361: if (p5 * rznorm*rznorm / delta2 <= rtol*(1.0-p5*rtol)*(par + rxnorm*rxnorm/delta2)) {
362: info = 1;
363: } else if (info == 0 && (p5*(par + rxnorm*rxnorm/delta2) <= atol/delta2)) {
364: info = 2;
365: }
366: }
368: /* Compute the Newton correction parc to par. */
369: if (xnorm == 0) {
370: parc = -par;
371: } else {
372: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, x, &blas1, wa2, &blas1));
373: temp = 1.0/xnorm;
374: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, wa2, &blas1));
375: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&blasn, &blas1, a, &blaslda, wa2, &blasn, &blasinfo));
376: temp = BLASnrm2_(&blasn, wa2, &blas1);
377: parc = (xnorm - delta)/(delta*temp*temp);
378: }
380: /* update parl or paru */
381: if (xnorm > delta) {
382: parl = PetscMax(parl, par);
383: } else if (xnorm < delta) {
384: paru = PetscMin(paru, par);
385: }
386: } else {
387: /* Case 2: A + par*I is not pos. def. */
389: /* Use the rank information from the Cholesky
390: decomposition to update par. */
392: if (indef > 1) {
393: /* Restore column indef to A + par*I. */
394: iblas = indef - 1;
395: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[indef-1 + 0*lda],&blaslda,&a[0 + (indef-1)*lda],&blas1));
396: a[indef-1 + (indef-1)*lda] = wa1[indef-1] + par;
398: /* compute parc. */
399: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[0 + (indef-1)*lda], &blas1, wa2, &blas1));
400: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&iblas,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
401: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,wa2,&blas1,&a[0 + (indef-1)*lda],&blas1));
402: temp = BLASnrm2_(&iblas,&a[0 + (indef-1)*lda],&blas1);
403: CHKMEMQ;
404: a[indef-1 + (indef-1)*lda] -= temp*temp;
405: PetscStackCallBLAS("LAPACKtrtr",LAPACKtrtrs_("U","N","N",&iblas,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
406: }
408: wa2[indef-1] = -1.0;
409: iblas = indef;
410: temp = BLASnrm2_(&iblas,wa2,&blas1);
411: parc = - a[indef-1 + (indef-1)*lda]/(temp*temp);
412: pars = PetscMax(pars,par+parc);
414: /* If necessary, increase paru slightly.
415: This is needed because in some exceptional situations
416: paru is the optimal value of par. */
418: paru = PetscMax(paru, (1.0+rtol)*pars);
419: }
421: /* Use pars to update parl */
422: parl = PetscMax(parl,pars);
424: /* Test for converged. */
425: if (info == 0) {
426: if (iter == itmax) info=4;
427: if (paru <= (1.0+p5*rtol)*pars) info=3;
428: if (paru == 0.0) info = 2;
429: }
431: /* If exiting, store the best approximation and restore
432: the upper triangle of A. */
434: if (info != 0) {
435: /* Compute the best current estimates for x and f. */
436: par = parf;
437: f = -p5 * (rxnorm*rxnorm + par*xnorm*xnorm);
438: if (rednc) {
439: f = -p5 * (rxnorm*rxnorm + par*delta*delta - rznorm*rznorm);
440: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn, &alpha, z, &blas1, x, &blas1));
441: }
442: /* Restore the upper triangle of A */
443: for (j = 0; j<n; j++) {
444: iblas = n - j - 1;
445: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j+1 + j*lda],&blas1, &a[j + (j+1)*lda],&blaslda));
446: }
447: iblas = lda+1;
448: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,wa1,&blas1,a,&iblas));
449: break;
450: }
451: par = PetscMax(parl,par+parc);
452: }
453: *retpar = par;
454: *retf = f;
455: *retinfo = info;
456: *retits = iter;
457: CHKMEMQ;
458: return(0);
459: }