diff --git a/docs/inverse/1955 markowitz.pdf b/docs/inverse/1955 markowitz.pdf
new file mode 100644
index 0000000..2702fdd
Binary files /dev/null and b/docs/inverse/1955 markowitz.pdf differ
diff --git a/pdp10/docs/algebra/ipwr.for b/pdp10/docs/algebra/ipwr.for
index d1f05ae..cca882b 100644
--- a/pdp10/docs/algebra/ipwr.for
+++ b/pdp10/docs/algebra/ipwr.for
@@ -1,11 +1,16 @@
-
-
+c imaginary powers of 10, starting with 10**(i/1024), and squaring successively ten times. 
+c this matches with feynman's table 22-3. 
       integer i
-      real pi
-      pi = 3.1415927
-      i = 0
-      i = i + 1
-10    format (f10.7)
-      write (*, 10) pi
+      real x, y, x2, y2
+10    format (2f10.5)
+      y = .00225
+      x = sqrt(1. - y**2)
+      do 20 i=1,11
+        write (6, 10) x, y
+        x2 = x**2 - y**2
+        y2 = 2 * x * y
+        x = x2
+        y = y2
+20    continue
       end
   
\ No newline at end of file
diff --git a/pdp10/docs/algebra/ipwr.mac b/pdp10/docs/algebra/ipwr.mac
new file mode 100644
index 0000000..a7011fd
--- /dev/null
+++ b/pdp10/docs/algebra/ipwr.mac
@@ -0,0 +1,68 @@
+a=1
+b=2
+c=3
+digit=4
+tmp=5
+sqr=6
+x=10
+y=11
+x2=12
+y2=13
+newlin: asciz   /
+/
+start:  reset
+        movei   y,^d225
+        fltr    y,y
+        fmpr    y,[0.00001]
+        move    a,y
+        fmp     a,a
+        movei   x,^d1
+        fltr    x,x
+        fsb     x,a
+        movei   sqr,1
+sqrloop:jsr     nxtsqr
+        addi    sqr,1
+        caig    sqr,13
+        jrst    sqrloop
+        exit
+nxtsqr: 0 ;next iteration on x and y
+        move    a,x
+        jsr     prflt
+        movei   tmp," "
+        outchr  tmp
+        move    a,y
+        jsr     prflt
+        outstr  newlin
+        move    x2,x
+        fmp     x2,x2
+        move    tmp,y
+        fmp     tmp,tmp
+        fsb     x2,tmp
+        move    y2,y
+        fmp     y2,x2
+        fmp     y2,[2.0]
+        move    x,x2
+        move    y,y2
+        jrstf   @nxtsqr
+prflt:  0 ;print float less than one in accum a
+        fix     b,a
+        fltr    c,b
+        fsbr    a,c
+        jumpge  b,point
+        movei   tmp,"-"
+        outchr  tmp
+point:  movei   tmp,"."
+        outchr  tmp
+        movei   digit,1
+digloop:fmpr    a,[10.0]
+        fix     tmp,a
+        fltr    c,tmp
+        fsbr    a,c
+        addi    tmp,"0"
+        outchr  tmp
+        addi    digit,1
+        caig    digit,5
+        jrst    digloop
+        jrstf   @prflt
+end start
+        
\ No newline at end of file
diff --git a/pdp10/docs/algebra/ipwr.py b/pdp10/docs/algebra/ipwr.py
index 6bf0b44..dff95ed 100644
--- a/pdp10/docs/algebra/ipwr.py
+++ b/pdp10/docs/algebra/ipwr.py
@@ -2,16 +2,6 @@
 feynman's table 22-3. the rounded version gives the same x value in the last iteration as feynman's, -0.66928. note 
 the initial y = .00225 should be .0022486, just as in the text. the same rounding is used here as there."""
 from math import sqrt
-
-print('rounded version')
-y = .00225
-x = round(sqrt(1. - y**2), 7)
-for _ in range(11):
-    print('%10.5f %10.5f' % (x, y))
-    x2 = round(x**2 - y**2, 7)
-    y2 = round(2 * x * y, 7)
-    x, y = x2, y2    
-print('\nnon rounded version')
 y = .00225
 x = sqrt(1. - y**2)
 for _ in range(11):
diff --git a/pdp10/docs/algebra/readme.md b/pdp10/docs/algebra/readme.md
new file mode 100644
index 0000000..edacf09
--- /dev/null
+++ b/pdp10/docs/algebra/readme.md
@@ -0,0 +1,233 @@
+target is using python, fortran10, macro10 in parallel to reproduce [feynman's table 22-3](https://www.feynmanlectures.caltech.edu/I_22.html)
+
+# skip to the punchline
+
+## python
+
+    from math import sqrt
+    y = .00225
+    x = sqrt(1. - y**2)
+    for _ in range(11):
+        print('%10.5f %10.5f' % (x, y))
+        x2 = x**2 - y**2
+        y2 = 2 * x * y
+        x, y = x2, y2
+
+## fortran10
+
+        integer i
+        real x, y, x2, y2
+    10  format (2f10.5)
+        y = .00225
+        x = sqrt(1. - y**2)
+        do 20 i=1,11
+            write (6, 10) x, y
+            x2 = x**2 - y**2
+            y2 = 2 * x * y
+            x = x2
+            y = y2
+    20  continue
+        end
+
+## macro10
+
+    a=1
+    b=2
+    c=3
+    digit=4
+    tmp=5
+    sqr=6
+    x=10
+    y=11
+    x2=12
+    y2=13
+    newlin: asciz   /
+    /
+    start:  reset
+            movei   y,^d225
+            fltr    y,y
+            fmpr    y,[0.00001]
+            move    a,y
+            fmp     a,a
+            movei   x,^d1
+            fltr    x,x
+            fsb     x,a
+            movei   sqr,1
+    sqrloop:jsr     nxtsqr
+            addi    sqr,1
+            caig    sqr,13
+            jrst    sqrloop
+            exit
+    nxtsqr: 0 ;next iteration on x and y
+            move    a,x
+            jsr     prflt
+            movei   tmp," "
+            outchr  tmp
+            move    a,y
+            jsr     prflt
+            outstr  newlin
+            move    x2,x
+            fmp     x2,x2
+            move    tmp,y
+            fmp     tmp,tmp
+            fsb     x2,tmp
+            move    y2,y
+            fmp     y2,x2
+            fmp     y2,[2.0]
+            move    x,x2
+            move    y,y2
+            jrstf   @nxtsqr
+    prflt:  0 ;print float less than one in accum a
+            fix     b,a
+            fltr    c,b
+            fsbr    a,c
+            jumpge  b,point
+            movei   tmp,"-"
+            outchr  tmp
+    point:  movei   tmp,"."
+            outchr  tmp
+            movei   digit,1
+    digloop:fmpr    a,[10.0]
+            fix     tmp,a
+            fltr    c,tmp
+            fsbr    a,c
+            addi    tmp,"0"
+            outchr  tmp
+            addi    digit,1
+            caig    digit,5
+            jrst    digloop
+            jrstf   @prflt
+    end start
+
+# stage1
+
+with fortran10, the basic workflow for iterating between raspi and tops10 [is there now](../sec5-minimalist-walkthrough.md). what would be nice is to be able to at the least do 'initial work' on fortran10's old school fortran iv/66 source code in a raspi ide debugger. clearly nothing like this will be possible for macro10, but for fortran10 it's worthwhile.
+
+onboard the raspi, insure that 'sudo apt install gcc' and 'sudo apt install gdb' are go. these cover the gfortran compiler, which does seem able to handle the source code. vscode and its ['modern fortran' extension](https://fortran-lang.org/) also work alright. for vscode run configurations, see [tasks.json](../../.vscode/tasks.json) and [launch.json](../../.vscode/launch.json).
+
+here's the [fortran10](ipwr.for). note in the write (6, 10) the '6' is a 'unit designation' and the '10' is a format statement line number. currently unit designation for terminal is 6 on raspi but 5 on tops10. would like to make this portable, no differences between raspi and tops10.
+
+with macro10, the 'print' of python and 'write' of fortran are the first topic. taking an accumulator containing a floating point number and printing it on the terminal in something like f10.5 format. the pdp10 has sixteen accumulators, and the numerical value contained in one of those is to be printed in 'human readable decimal form' on the terminal. 
+
+before that, first step is to do the same but for an accumulator containing a fixed point twos complement binary number. the historical 'decimal output / decout' problem, as discussed in the 'early sixties' section of the levy book, and a kind of historical landmark, along with the contemporary topics of recursion, stack processing, algol, and the beginnings of academic computer science.
+
+    a=1
+    b=2
+    p=17
+    pdlen==40
+    pdlist: block   pdlen
+    opdef   call    [pushj p,]
+    opdef   ret     [popj p,]
+    crlf:   byte    (7)15,12
+    start:  reset
+            move    p,[iowd pdlen,pdlist]
+            movei   a,3
+            call    decout
+            hrroi   a,crlf
+            outstr  (a)
+            exit
+    decout: jumpge  a,decot1
+            push    p,a
+            movei   a,"-"
+            outchr  a
+            pop     p,a
+            movn    a,a
+    decot1: idivi   a,^d10
+            push    p,b
+            skipe   a
+            call    decot1
+            pop     p,a
+            addi    a,"0"
+            outchr  a
+            ret
+    end start
+
+# stage2
+
+printing floating point is a modification of the above, bringing in the pdp10 floating point instructions. there's a numerical value to be printed out to the terminal, and that's done one character at a time using the 'outchr' tops10 muuo. this is a request to tops10 to print a character on the terminal, and it's used repeatedly to print each character of the numerical value, including negative sign and decimal point.
+
+as part of understanding all of this, it became apparent that a stack isn't needed. the code above is from the gorin book and it's oriented towards academic computer science. here's a minimalist numerical computing approach. for feynman's table 22-3, the need is to print five digits after the decimal point. that's all this code does. at the start, accumulator 'a' contains a float that is less than one. it's 'after the decimal point'. the machinery repeats five times, cranking five digits out 'in front of the decimal point', one by one, to print them with 'outchr'.
+
+            movei   tmp,"."
+            outchr  tmp   
+            movei   itr,1
+    loop:   fmpr    a,[10.0]
+            fix     tmp,a
+            fltr    c,tmp
+            fsbr    a,c
+            addi    tmp,"0"
+            outchr  tmp
+            addi    itr,1
+            caig    itr,5
+            jrst    loop
+            ...
+
+the code below works. it reproduces feynman's table 22-3 and the results of the python and fortran10 code. something interesting happens though, from about the sixth or seventh of the eleven iterations. this is actually an opportunity to learn more about the early days of floating point and its practical use, so will be tackled deliberately going forward. the code below is using standard single precision pdp10 floating point, so 27 bits of precision. the first diagnosis has to be that using double precision would cure the problem. if this turns out to be true, it will be an excellent example of real-world effects of floating point precision, and will mean that feynman's table 22-3 is an excellent detector of poor numerical precision. 
+
+    a=1
+    b=2
+    c=3
+    digit=4
+    tmp=5
+    sqr=6
+    x=10
+    y=11
+    x2=12
+    y2=13
+    newlin: asciz   /
+    /
+    start:  reset
+            movei   y,^d225
+            fltr    y,y
+            fmpr    y,[0.00001]
+            move    a,y
+            fmp     a,a
+            movei   x,^d1
+            fltr    x,x
+            fsb     x,a
+            movei   sqr,1
+    sqrloop:jsr     nxtsqr
+            addi    sqr,1
+            caig    sqr,13
+            jrst    sqrloop
+            exit
+    nxtsqr: 0 ;next iteration on x and y
+            move    a,x
+            jsr     prflt
+            movei   tmp," "
+            outchr  tmp
+            move    a,y
+            jsr     prflt
+            outstr  newlin
+            move    x2,x
+            fmp     x2,x2
+            move    tmp,y
+            fmp     tmp,tmp
+            fsb     x2,tmp
+            move    y2,y
+            fmp     y2,x2
+            fmp     y2,[2.0]
+            move    x,x2
+            move    y,y2
+            jrstf   @nxtsqr
+    prflt:  0 ;print float less than one in accum a
+            fix     b,a
+            fltr    c,b
+            fsbr    a,c
+            jumpge  b,point
+            movei   tmp,"-"
+            outchr  tmp
+    point:  movei   tmp,"."
+            outchr  tmp
+            movei   digit,1
+    digloop:fmpr    a,[10.0]
+            fix     tmp,a
+            fltr    c,tmp
+            fsbr    a,c
+            addi    tmp,"0"
+            outchr  tmp
+            addi    digit,1
+            caig    digit,5
+            jrst    digloop
+            jrstf   @prflt
+    end start