fix derivations in comments
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rlar
Holger Vogt
parent
af351eab5f
commit
e36ded8122
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@ -296,12 +296,28 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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break;
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case PT_POWER:
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/*
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* ^ : a^b -> |a| math^ b
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*
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* D(pow(a,b))
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* = D(exp(b*log(abs(a))))
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* = exp(b*log(abs(a))) * D(b*log(abs(a)))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*D(abs(a))/abs(a))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*sgn(a)*D(a)/abs(a))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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*
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* when D(b) == 0, then
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*
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* D(pow(a,b))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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* = pow(a,b) * b * D(a)/a
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* = pow(a,b) * b * D(a)/(signum(a) * abs(a))
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* = pow(a, b-1) * b * D(a) / signum(a)
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* = pwr(a, b-1) * b * D(a)
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*/
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#define a p->left
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#define b p->right
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if (b->type == PT_CONSTANT) {
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/*
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* D(a^C) = C * a^(C-1) * D(a)
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*/
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arg1 = PTdifferentiate(a, varnum);
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newp = mkb(PT_TIMES, mkb(PT_TIMES,
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@ -310,11 +326,6 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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mkcon(b->constant - 1))),
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arg1);
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} else {
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/*
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* D(a^b) = D(exp(b*log(a)))
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* = exp(b*log(a)) * D(b*log(a))
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* = exp(b*log(a)) * (D(b)*log(a) + b*D(a)/a)
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*/
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arg1 = PTdifferentiate(a, varnum);
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arg2 = PTdifferentiate(b, varnum);
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newp = mkb(PT_TIMES, mkf(PTF_EXP, mkb(PT_TIMES,
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@ -551,6 +562,25 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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break;
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case PTF_POW:
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/*
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* pow : pow(a,b) -> |a| math^ b
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*
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* D(pow(a,b))
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* = D(exp(b*log(abs(a))))
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* = exp(b*log(abs(a))) * D(b*log(abs(a)))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*D(abs(a))/abs(a))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*sgn(a)*D(a)/abs(a))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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*
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* when D(b) == 0, then
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*
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* D(pow(a,b))
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* = pow(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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* = pow(a,b) * b * D(a)/a
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* = pow(a,b) * b * D(a)/(signum(a) * abs(a))
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* = pow(a, b-1) * b * D(a) / signum(a)
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* = pwr(a, b-1) * b * D(a)
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*/
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{
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/*
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pow(a,b)
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@ -560,9 +590,6 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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#define b p->left->right
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if (b->type == PT_CONSTANT) {
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/*
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* D(f^C) = C * f^(C-1) * D(f)
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*/
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arg1 = PTdifferentiate(a, varnum);
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newp = mkb(PT_TIMES, mkb(PT_TIMES,
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@ -576,11 +603,6 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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printf("\n");
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#endif
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} else {
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/*
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* D(f^g) = D(exp(g*log(f)))
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* = exp(g*log(f)) * D(g*log(f))
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* = exp(g*log(f)) * (D(g)*log(f) + g*D(f)/f)
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*/
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arg1 = PTdifferentiate(a, varnum);
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arg2 = PTdifferentiate(b, varnum);
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newp = mkb(PT_TIMES, mkf(PTF_EXP, mkb(PT_TIMES,
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@ -599,6 +621,26 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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break;
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case PTF_PWR:
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/*
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* pwr : pwr(a,b) -> signum(a) * (|a| math^ b)
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* -> signum(a) * pow(a, b)
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*
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* Note:
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* D(pow(a,b)) = pow(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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*
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* D(pwr(a,b))
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* = D(signum(a) * pow(a,b))
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* = D(signum(a)) * pow(a,b) + signum(a) * D(pow(a,b))
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* = 0 + signum(a) * pow(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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* = pwr(a,b) * (D(b)*log(abs(a)) + b*D(a)/a)
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*
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* with D(b) == 0
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*
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* D(pwr(a,b))
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* = pwr(a,b) * b * D(a)/a
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* = signum(a) * pow(a,b) * b * D(a)/(signum(a) * abs(a))
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* = pow(a, b-1) * b * D(a)
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*/
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{
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/*
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pwr(a,b)
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@ -607,18 +649,6 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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#define a p->left->left
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#define b p->left->right
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if (b->type == PT_CONSTANT) {
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/* b is a constant
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*
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* f(a,b) = signum(a) * abs(a)^b
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* = signum(a) * exp(b*log(abs(a)))
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* D(f) = signum(a) * D(exp(b*log(abs(a))))
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* = signum(a) * exp(b*log(abs(a))) * D(b*log(abs(a)))
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* = signum(a) * abs(a)^b * D(b*log(abs(a)))
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* = signum(a) * abs(a)^b * b * 1/abs(a) * D(abs(a))
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* = signum(a) * abs(a)^(b-1) * b * D(abs(a))
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* = signum(a) * abs(a)^(b-1) * b * signum(a) * D(a)
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* = abs(a)^(b-1) * b * D(a)
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*/
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arg1 = PTdifferentiate(a, varnum);
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newp = mkb(PT_TIMES,
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@ -634,19 +664,6 @@ static INPparseNode *PTdifferentiate(INPparseNode * p, int varnum)
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printf("\n");
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#endif
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} else {
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/* b is a function
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*
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* f(a,b) = signum(a) * abs(a)^b
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* = signum(a) * exp(b*log(abs(a)))
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* D(f) = signum(a) * D(exp(b*log(abs(a))))
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* = signum(a) * exp(b*log(abs(a))) * D(b*log(abs(a)))
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* = signum(a) * exp(b*log(abs(a))) * (D(b) * log(abs(a)) + b * D(log(abs(a))))
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* = signum(a) * exp(b*log(abs(a))) * (D(b) * log(abs(a)) + b * 1/abs(a) * D(abs(a)))
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* = signum(a) * exp(b*log(abs(a))) * (D(b) * log(abs(a)) + b * 1/abs(a) * signum(a)*D(a))
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* = signum(a) * exp(b*log(abs(a))) * (D(b) * log(abs(a)) + b/a*D(a))
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* = signum(a) * exp(b*log(abs(a))) * D(b) * log(abs(a) + signum(a) * exp(b*log(abs(a))) / a * b * D(a)
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* = signum(a) * exp(b*log(abs(a))) * D(b) * log(abs(a) + abs(a)^(b-1) * b * D(a)
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*/
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arg1 = PTdifferentiate(a, varnum);
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arg2 = PTdifferentiate(b, varnum);
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newp = mkb(PT_PLUS,
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